Research ArticleConstruction of Quasi-Cyclic LDPC Codes Based onFundamental Theorem of Arithmetic
Hai Zhu 1 Liqun Pu2 Hengzhou Xu 1 and Bo Zhang1
1School of Network Engineering Zhoukou Normal University Zhoukou China2School of Mathematics and Statistics Zhengzhou University Zhengzhou China
Correspondence should be addressed to Hai Zhu zhu sea163com and Hengzhou Xu hzxuzknueducn
Received 23 November 2017 Revised 5 February 2018 Accepted 7 March 2018 Published 15 April 2018
Academic Editor Qin Huang
Copyright copy 2018 Hai Zhu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Quasi-cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for5G enhanced mobile broadband (eMBB) data channel In this paper we study the construction of QC LDPC codes based on anarbitrary given expansion factor (or lifting degree) First we analyze the cycle structure of QC LDPC codes and give the necessaryand sufficient condition for the existence of short cycles Based on the fundamental theorem of arithmetic in number theory wedivide the integer factorization into three cases and present three classes of QC LDPC codes accordingly Furthermore a generalconstructionmethod of QC LDPC codes with girth of at least 6 is proposed Numerical results show that the constructed QC LDPCcodes perform well over the AWGN channel when decoded with the iterative algorithms
1 Introduction
Low-density parity-check (LDPC) codes [1] are a class ofmodern channel coding Because of the advantages of ap-proaching the Shannon capacity and the iterative decodingalgorithms with lower complexity LDPC codes have beenattracting great interests of the industries and academiaFor various specific communication systems [2ndash4] LDPCcodes have been well designed and chosen as their standardcodes As an important scenario of 5G communicationsthe enhanced mobile broadband (eMBB) data channel hadadopted the LDPC coding scheme [5] and LDPC codes haverecently been determined after several rounds of discussions[6ndash12] However the other two scenarios of 5G commu-nications that is ultrareliable and low latency communi-cations (URLLC) and massive machine-type-communication(mMTC) have no candidate channel coding at present Thepromising coding techniques for 5G communication systemsare turbo codes binarynonbinary LDPC codes spatiallycoupled (SC) LDPC codes [13] block Markov superpositiontransmission (BMST) [14] and polar codes The encod-ingdecoding complexity performance spectral efficiencyand robustness comparisons among them can be found in[15] Recently some low-complexity decoding algorithms
of these modern channel codes have been proposed [1617] These significant works can facilitate and acceleratethe applications of these modern coding techniques in 5Gcommunications According to the definition and descriptionof URLLC and mMTC provided by ITU-R [18] these twoscenarios require low latency and high reliability That isshort data package communicationwhich has no visible errorfloor down to block error rate (BLER) of 10minus5 should beconsidered Research results [19] show that LDPC codes havegood performance in the waterfall and error-floor regionMoreover LDPC codes have good robust property [15 20]and then their good performance can be also obtained overvarious channels Hence LDPC coding still has a strongcompetitiveness in the applications of URLLC and mMTC
LDPC codes can be divided into two major classes(1) random-like codes constructed by means of computersearch under the efficient algorithms [21 22] and (2) struc-tured codes constructed based on algebraic tools combi-natorial structures and graphs such as finite geometries[23] finite fields [24] balanced incomplete block designs(BIBDs) [20] resolvable group divisible designs (RGDDs)[25] and protographs [26 27] Research results show thatwell designed algebraic-based LDPC codes have no errorfloor at the bit error rate (BER) down to 10minus15 [28] To
HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 5264724 9 pageshttpsdoiorg10115520185264724
2 Wireless Communications and Mobile Computing
facilitate implementation LDPC codes usually have somespecial structures such as diagonal structure and quasi-cyclic(QC) structure In general quasi-cyclic (QC) LDPC codes[29] have advantages of encoding and decoding with lowcomplexity [30 31] easy hardware implementation [32] andgood iterative performance [33] and then they have attractedcomprehensive attention
In order to support lots of data packets with variouslengths in the eMBB scenario of 5G communications thedesigned 5G LDPC codes are chosen as rate-compatible (RC)QCLDPC codes Notice that the number of expansion factors(or lifting degrees) of 5GQCLDPC codes is notmuchOn theother hand some encoding algorithms [34] are only suitablefor QC LDPC codes with certain expansion factor (or liftingdegree) Furthermore the encoding and decoding of QCLDPC codes with expansion factors (or lifting degrees) beingthe power of two can be easily implemented by linear shiftregisters Hence it is interesting to construct QCLDPC codesfrom an arbitrary given expansion factor (or lifting degree)
In this paper we focus on the construction of QC LDPCcodes from given expansion factors (or lifting degrees) Wefirst introduce the fundamental theorem of arithmetic innumber theory and divide the integer factorization into threecategories By analyzing the cycle structure of QC LDPCcodes we present three classes of QC LDPC codes based onthree families of integers Furthermore a general construc-tion of QC LDPC codes with girth of at least 6 based on thefundamental theorem of arithmetic is proposed Finally inorder to show the good performance of our constructed QCLDPC codes numerical simulation results are provided
The rest of this paper is organized as follows Section 2introduces the fundamentals of number theory the defini-tions basic concepts and cycle structure of QC LDPC codesSection 3 presents three classes of QC LDPC codes and ageneral constructionmethod Numerical results are also pro-vided in this section Finally Section 4 concludes this paper
2 Preliminaries
21 Fundamentals of Number Theory
Theorem 1 Every composite number which is greater thanone factors uniquely as a product of prime numbers
This theorem is the well-known fundamental theoremof arithmetic in number theory and it had been proved byGauss and Clarke [36]This theorem is also called the uniquefactorization theoremThat is every integer greater than oneis either prime itself or the product of the prime numbersand this product is unique up to the order of the factors Forexample 1400 = 14times 100 = 2times 7times 2times 2times 5times 5 = 23 times 52 times 7This theorem is twofold first 1400 can bewritten as a productof the primes and second no matter how this is done therewill always be three 2s two 5s one 7 and no other primesin this product Hence for a given integer 119871 ge 2 119871 can berepresented by the unique product that is
119871 = 11990111989011 times 11990111989022 times sdot sdot sdot times 119901119890119896119896 (1)
where 1199011 1199012 119901119896 are prime numbers
The following three lemmas and one theorem are usefulfor constructing QC LDPC codes with girth of at least 6
Lemma 2 Let 119901 be a prime and let 1198901 and 1198902 be two positiveintegers If 119886 and 119887 are two integers for 1 le 119886 le 1199011198901 minus 1 and1 le 119887 le 1199011198902 minus 1 then 119886119887 = 0 (mod1199011198901+1198902)Proof Since 119886 and 119887 are two integers with 1 le 119886 le 1199011198901 minus1 and1 le 119887 le 1199011198902 minus 1 then
1 = 1 times 1 le 119886 times 119887 le (1199011198901 minus 1) (1199011198902 minus 1) lt 1199011198901+1198902 (2)
That is 119886119887 = 0 (mod1199011198901+1198902)Lemma 3 Let 1199011 and 1199012 be two different primes and let 1198901and 1198902 be two positive integers If 119886 and 119887 are two integers for1 le 119886 le 11990111989011 minus1 and 1 le 119887 le 11990111989022 minus1 then 119886119887 = 0 (mod11990111989011 11990111989022 )Proof Since 119886 and 119887 are two integers with 1 le 119886 le 11990111989011 minus1 and1 le 119887 le 11990111989022 minus 1 then
1 le 119886 times 119887 le (11990111989011 minus 1) (11990111989022 minus 1) lt 11990111989011 11990111989022 (3)
That is 119886119887 = 0 (mod11990111989011 11990111989022 )Lemma 4 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 be a positive integer where1199011 1199012 119901119896 are 119896 different primes and 1198901 1198902 119890119896 are 119896positive integers If 119886 = 119901119890119894119894 and 119887 = 119901119890119895119895 with 1 le 119894 119895 le 119896and 119894 = 119895 then 119886119887 = 0 (mod119871)Proof Since 119886 = 119901119890119894119894 and 119887 = 119901119890119895119895 with 1 le 119894 119895 le 119896 and 119894 = 119895then
1 le 119886 times 119887 le 119901119890119894119894 119901119890119895119895 lt 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 = 119871 (4)
That is 119886119887 = 0 (mod119871)Theorem 5 Let 119871 be a positive integer If 119886 and 119887 are twopositive integers and 119886119887 lt 119871 then 119886119887 = 0 (mod119871)Proof Since 119886 and 119887 are two positive integers and 119886119887 lt 119871then
1 le 119886119887 lt 119871 (5)
That is 119886119887 = 0 (mod119871)22 QC LDPC Codes and Their Associated Tanner Graphs A(120574 120588)-regular quasi-cyclic (QC) LDPC code [29] of length 120588119871can be completely specified by the null space of the followingmatrix over GF(2)
H =[[[[[[[[
I (11990100) I (11990101) sdot sdot sdot I (1199010120588minus1)I (11990110) I (11990111) sdot sdot sdot I (1199011120588minus1)
d
I (119901120574minus10) I (119901120574minus11) sdot sdot sdot I (119901120574minus1120588minus1)
]]]]]]]]
(6)
where for 0 le 119894 le 120574 minus 1 and 0 le 119895 le 120588 minus 1 I(119901119894119895) is an 119871 times 119871circulant permutation matrix (CPM) with a one at column-(119903+119901119894119895) (mod119871) for row-119903 0 le 119903 le 119871minus1 and zero elsewhere
Wireless Communications and Mobile Computing 3
H =
[[[[[[[
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
]]]]]]]
(a)
Checknodes
6-cycle
Variablenodes
0
0
1
1
2
2
3
3 4 5(b)
Figure 1 Tanner graph ofH
It is clear that I(0) represents the 119871times119871 identity matrix Noticethat the parameter 119871 is referred to as the expansion factor (orlifting degree) [37] It can be easily observed that the positionsof nonzero elements in H are uniquely determined by thefollowing matrix called permutation shift matrix or exponentmatrix
That is there is a one-to-one correspondence between P andH
An LDPC code is commonly described by a bipartitegraph known as Tanner graph [38] in coding theory Tannergraph ofH denoted byG(119881 119862) consists of a set119881 of variablenodes (containing 120588119871 code symbols of a code word) and a set119862 of check nodes (containing 120574119871 local check-sum constraintson the code symbols) An edge in G(119881 119862) connects thevariable node 119894 to the check node 119895 if and only if the elementat column-119894 and row-119895 of H is nonzero A cycle is formedby a sequence of vertices (or edges) in G(119881 119862) which startsand ends at the same vertex (or edge) and contains othervertices (or edges) not more than once The cycle of length119896 is denoted as 119896-cycle for short and the length of the shortestcycle is called the girth of G(119881 119862) (or an LDPC code) Asan example Figure 1 shows the Tanner graph of H and anassociate 6-cycle
In graph theory the biadjacency matrix A = [119886119894119895] of abipartite graph G(119880 119881) can be constructed as follows Therows ofA are labeled by the |119880| vertices in119880 and the columnsare labeled by |119881| vertices in 119881 The element 119886119894119895 at the rowlabeled by the vertex 119894 isin 119880 and the column labeled by thevertex 119895 isin 119881 is 1 if and only if there exists an edge between thevertices 119894 and 119895 and otherwise 0 Actually for an LDPC codegiven by the null space of H H is the biadjacency matrix ofits relevant Tanner graphG(119881 119862)
Moreover isomorphism theory of QC LDPC codes wasproposed in [39ndash41] based on the isomorphism of graphs ingraph theory According to the isomorphism of QC LDPC
codes the parity-check matrix in (6) can be simplified as thefollowing matrix
H =[[[[[[[[
I (0) I (0) sdot sdot sdot I (0)I (0) I (11990111) sdot sdot sdot I (1199011120588minus1) d
I (0) I (119901120574minus11) sdot sdot sdot I (119901120574minus1120588minus1)
]]]]]]]]
(8)
That is 1199011198940 = 1199010119895 = 0 for 0 le 119894 le 120574 minus 1 0 le 119895 le 120588 minus 1Equivalently its exponent matrix is
That is why the elements in the first row and first columnof the exponent matrix P are usually set to 0 in the researchprocess [29 42] Hence we only consider suchH and P in thefollowing discussions
23 Cycle Structure ofQCLDPCCodes Consider aQCLDPCcodeC given by the null space ofH in (8) It can be seen from[29] that a cycle in the Tanner graph ofC is associated with afamily of the ordered CPMs inH As shown in [29] a 2119894-cyclein the Tanner graph of the codeC (orH) is represented by anordered sequence of CPMs
I (1199011198950 1198960) I (1199011198951 1198960) I (1199011198951 1198961) I (1199011198952 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) I (1199011198950 119896119894minus1) I (1199011198950 1198960)
(10)
where 119895119894 = 1198950 119896119894 = 1198960 0 le 119895119898 le 120574minus 1 119895119898minus1 = 119895119898 0 le 119896119898 le120588 minus 1 and 119896119898minus1 = 119896119898 for 1 le 119898 le 119894 The above sequence canbe simplified as
I (1199011198950 1198960) I (1199011198951 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) (11)
It can be seen that such a 2119894-cycle corresponds to the elements1199011198950 1198960 1199011198951 1198961 1199011198952 1198962 119901119895119894minus1 119896119894minus1 in the exponent matrix P Fur-thermore short cycles of QC LDPC codes can be determinedby the elements of P [39 40]
4 Wireless Communications and Mobile Computing
(pjk) (pj0k2minus1
)
(pj+1k+1)
(pj2minus1k2minus1)
(pj+1k)
pj0k0
pj1k0pj1k1
pj0kminus1
pjminus1kminus1
Figure 2The structure of 2119894-cycle and nonexistence of the 4119894-cycle
Let 119892 be the girth of the codeC It can be seen from [43]that for119892 le 2119894 le 2119892minus2 the necessary and sufficient conditionfor the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) can be generalized as follows
119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = 0 (mod 119871) (12)
with 1198950 = 119895119894 1198960 = 119896119894 119896119898 = 119896119898+1 and 119895119898 = 119895119898+1 Note that(12) is not the sufficient condition for the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) for 2119894 ge 2119892 butit is the necessary condition Here we give a counterexampleConsider a 2119894-cycle (119894 ge 119892) whose cycle structure is given inFigure 2 Clearly (12) is satisfied Let 119901119895119898 119896119898 = 119901119895119898+119894119896119898+119894 for0 le 119898 le 119894 minus 1 and 1198950 = 1198952119894 1198960 = 1198962119894 119896119898 = 119896119898+1 119895119898 = 119895119898+1According to (12) we have
2119894minus1
sum119899=0
(119901119895119899 119896119899 minus 119901119895119899+1 119896119899) = 2119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1119896119898)
= 0 (mod119871) (13)
where 1198950 = 119895119894 = 1198952119894 1198960 = 119896119894 = 1198962119894 119896119899 = 119896119899+1 119895119899 =119895119899+1 for 0 le 119899 le 2119894 minus 1 That is the ordered elements1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 1198962119894minus1 (ie 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1) make (12) hold butthey do not determine a 4119894-cycle A visual representation isdepicted in Figure 2 Therefore (4) in [29] and (3) in [39] arenot applicable to the cycles with lengths larger than 2119892 minus 2
3 Construction of Quasi-Cyclic LDPC Codeswith Girth of at Least 6
Based on the aforementioned there exists a one-to-one cor-respondence between the exponent matrix P and the parity-check matrix H of a QC LDPC code Hence construction ofa QC LDPC code is equivalent to the design of its exponentmatrix P In this section we present three classes of QCLDPC codes with girth of at least 6 and then give a generalconstruction of QC LDPC codes with girth of at least 6 basedon an arbitrary integer
First we design the exponent matrix P in (9) as follows
where 119901119894119895 = 119894 times 119895 (mod119871) for 1 le 119894 le 120574 minus 1 1 le 119895 le 120588 minus 1Second we replace the 0s and 119901119894119895 in the designed exponentmatrix P with CPMs I(0) and I(119901119894119895) of the same size 119871 times 119871respectively and then obtain a 120574 times 120588 arrayH of 119871 times 119871 CPMsThis array is a 120574119871 times 120588119871 matrix over GF(2) with column androwweights 120574 and120588 respectivelyThenull space of thismatrixgives a (120574 120588)-regular QC LDPC code
Remark 6 As shown in [44] girth and short cycles play animportant role in the design of LDPC codes If the aboveconstructed (120574 120588)-regular QC LDPC code does not havegood iterative performance we can replace someCPMs in theabove array H with zero matrices (ZMs) of the same size toreduce the number of short cycles and possibly enlarge thegirth value This replacement is called masking On the otherhand if the lengths of the desiredQCLDPC codes are shorterthan 120588119871 or they require much higher code rates then we cantake a 1205741015840 times 1205881015840 subarray of the designed arrayH where 1205741015840 le 120574and 1205881015840 le 120588 Notice that this subarray can be obtained fromthe following two steps (1) Choose the first 1205741015840 row-CPMsof the designed array H (2) select 1205881015840 column-CPMs from 120588column-CPMs of the designed array H In this paper boththe masking technique and the selection method in [43] areemployed to construct (or further optimize)QCLDPCcodes
31 Three Classes of QC LDPC Codes with Girth of at Least 6Based on (12) we can see that Tanner graph of the designedarrayH contains a 4-cycle if and only if the following equationis satisfied
1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = (1198950 minus 1198951) (1198960 minus 1198961)= 0 (mod119871)
(15)
where 1198950 = 1198951 and 1198960 = 1198961 It can be observed that theexistence of 4-cycles in theTanner graph of the designed arrayH is related to 119871 According to the fundamental theorem ofarithmetic the values of 119871 can be divided into three categoriesand three classes of QC LDPC codes with girth of at least6 are proposed Notice that all numerical simulations in thefollowing examples binary phase shift keying (BPSK) additivewhite Gaussian noise (AWGN) channel and the sum-productalgorithm (SPA) are assumed
311 The Case of 119871 = 119901119890 Let 119871 = 119901119890 where 119901 is a primeand 119890 is a positive integer and let 119890 = 1198901 + 1198902 where 1198901 1198902are two positive integers and 1198901 le 1198902 Consider 120574 = 1199011198901 and120588 = 1199011198902 Since 1 le 1198950 1198951 le 1199011198901 minus 1 1 le 1198960 1198961 le 1199011198902 minus1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 1199011198901 minus 1 and1 le 1198960 minus 1198961 le 1199011198902 minus 1 where the calculation is taken modulo1199011198901 and modulo 1199011198902 respectively Hence (15) is not satisfied
Wireless Communications and Mobile Computing 5
according to Lemma 2 That is Tanner graph of the designedarray H has no 4-cycles and then the constructed QC LDPCcodes have girth of at least 6
Example 7 Consider 119871 = 256 = 28 Let 120574 = 22 and 120588 = 26According to (14) we can obtain the exponentmatrixP of size4 times 64 By employing the method in [43] we select the first 4rows and the 2nd 16th 19th 35th 50th 55th 62nd and 63rdcolumns of P and construct a 4times8 arrayH of 256times256CPMsby replacing the elements of the selected submatrix with thecorresponding CPMs By using the matrix
to mask H a 1024 times 2048 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (2048 1024) QC LDPC code The bit errorrates (BERs) of this code decoded by the SPA (5 10 20 and 50iterations) are shown in Figure 3 Also shown in Figure 3 is theperformance of the (3 6)-regular (2048 1024) algebraic QCLDPC code constructed based on finite fieldGF(119901119904) [35]Thiscomparable code is constructed from the prime field GF(257)and then the CPM size of its parity-check matrix is 256times256Notice that the exponent matrix and masking matrix of thiscomparable code are
respectively It can be observed that these two codes havesimilar performance when decoded using the SPA withvarious iterations It is well known that algebraic-based LDPCcodes have fast decoding convergence [19 45 46]That is theSPAdecoding of the proposed LDPC code also converges fastas shown in Figure 3 We can see that the performance gapbetween 20 and 50 iterations is less than 015 dB at the BER of10minus6 and the gap is also less than 025 dB at the BER of 10minus7hence this code achieves a fast rate of decoding convergence
312 The Case of 119871 = 11990111989011 11990111989022 Let 119871 = 11990111989011 11990111989022 where 1199011 1199012are two different prime numbers and 1198901 1198902 are two positiveintegers Assume 120574 = min11990111989011 11990111989022 and 120588 = max11990111989011 11990111989022 Without loss of generality 11990111989011 lt 11990111989022 is assumed Since 1 le1198950 1198951 le 11990111989011 minus1 1 le 1198960 1198961 le 11990111989022 minus1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950minus1198951 le 11990111989011 minus1 and 1 le 1198960minus1198961 le 11990111989022 minus1 where the calcula-tion is takenmodulo11990111989011 andmodulo11990111989022 respectively Hence(15) is not satisfied according to Lemma 3 That is Tannergraph of the designed array H does not contain 4-cycles andthe girth of the constructed QC LDPC codes is at least 6
Figure 3 The bit error performance of the proposed (3 6)-regular(2048 1024) QC LDPC code and the comparable (3 6)-regular(2048 1024) algebraic QC LDPC code [19] in Example 7 Thedecoding algorithm is the SPA with 5 10 20 and 50 iterations
Example 8 Consider 119871 = 72 = 23times32 = 4times18 Since 12 lt 18let 120574 = 22 and 120588 = 18 According to (14) we can obtain theexponent matrix P of size 4times18 By employing themethod in[43] we select the first 4 rows and the 1st 2nd 3rd 4th 5th6th 12th 13th 15th 16th 17th and 18th columns of P andconstruct a 4 times 12 array H of 72 times 72 CPMs by replacing theelements of the selected submatrix with the correspondingCPMs By using the matrix
to maskH a 288 times 864matrix with column and row weights3 and 9 respectively is obtained This matrix gives a (3 9)-regular (864 576) QC LDPC code The bitword error rates(BERsWERs) of this code decoded by the SPA with 50iterations are shown in Figure 4 Also shown in Figure 4 isthe performance of the (3 9)-regular (864 576) algebraic QCLDPC code constructed from the finite field GF(73) [35]Theexponent and masking matrices of this algebraic QC LDPCcode are
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
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Submit your manuscripts atwwwhindawicom
2 Wireless Communications and Mobile Computing
facilitate implementation LDPC codes usually have somespecial structures such as diagonal structure and quasi-cyclic(QC) structure In general quasi-cyclic (QC) LDPC codes[29] have advantages of encoding and decoding with lowcomplexity [30 31] easy hardware implementation [32] andgood iterative performance [33] and then they have attractedcomprehensive attention
In order to support lots of data packets with variouslengths in the eMBB scenario of 5G communications thedesigned 5G LDPC codes are chosen as rate-compatible (RC)QCLDPC codes Notice that the number of expansion factors(or lifting degrees) of 5GQCLDPC codes is notmuchOn theother hand some encoding algorithms [34] are only suitablefor QC LDPC codes with certain expansion factor (or liftingdegree) Furthermore the encoding and decoding of QCLDPC codes with expansion factors (or lifting degrees) beingthe power of two can be easily implemented by linear shiftregisters Hence it is interesting to construct QCLDPC codesfrom an arbitrary given expansion factor (or lifting degree)
In this paper we focus on the construction of QC LDPCcodes from given expansion factors (or lifting degrees) Wefirst introduce the fundamental theorem of arithmetic innumber theory and divide the integer factorization into threecategories By analyzing the cycle structure of QC LDPCcodes we present three classes of QC LDPC codes based onthree families of integers Furthermore a general construc-tion of QC LDPC codes with girth of at least 6 based on thefundamental theorem of arithmetic is proposed Finally inorder to show the good performance of our constructed QCLDPC codes numerical simulation results are provided
The rest of this paper is organized as follows Section 2introduces the fundamentals of number theory the defini-tions basic concepts and cycle structure of QC LDPC codesSection 3 presents three classes of QC LDPC codes and ageneral constructionmethod Numerical results are also pro-vided in this section Finally Section 4 concludes this paper
2 Preliminaries
21 Fundamentals of Number Theory
Theorem 1 Every composite number which is greater thanone factors uniquely as a product of prime numbers
This theorem is the well-known fundamental theoremof arithmetic in number theory and it had been proved byGauss and Clarke [36]This theorem is also called the uniquefactorization theoremThat is every integer greater than oneis either prime itself or the product of the prime numbersand this product is unique up to the order of the factors Forexample 1400 = 14times 100 = 2times 7times 2times 2times 5times 5 = 23 times 52 times 7This theorem is twofold first 1400 can bewritten as a productof the primes and second no matter how this is done therewill always be three 2s two 5s one 7 and no other primesin this product Hence for a given integer 119871 ge 2 119871 can berepresented by the unique product that is
119871 = 11990111989011 times 11990111989022 times sdot sdot sdot times 119901119890119896119896 (1)
where 1199011 1199012 119901119896 are prime numbers
The following three lemmas and one theorem are usefulfor constructing QC LDPC codes with girth of at least 6
Lemma 2 Let 119901 be a prime and let 1198901 and 1198902 be two positiveintegers If 119886 and 119887 are two integers for 1 le 119886 le 1199011198901 minus 1 and1 le 119887 le 1199011198902 minus 1 then 119886119887 = 0 (mod1199011198901+1198902)Proof Since 119886 and 119887 are two integers with 1 le 119886 le 1199011198901 minus1 and1 le 119887 le 1199011198902 minus 1 then
1 = 1 times 1 le 119886 times 119887 le (1199011198901 minus 1) (1199011198902 minus 1) lt 1199011198901+1198902 (2)
That is 119886119887 = 0 (mod1199011198901+1198902)Lemma 3 Let 1199011 and 1199012 be two different primes and let 1198901and 1198902 be two positive integers If 119886 and 119887 are two integers for1 le 119886 le 11990111989011 minus1 and 1 le 119887 le 11990111989022 minus1 then 119886119887 = 0 (mod11990111989011 11990111989022 )Proof Since 119886 and 119887 are two integers with 1 le 119886 le 11990111989011 minus1 and1 le 119887 le 11990111989022 minus 1 then
1 le 119886 times 119887 le (11990111989011 minus 1) (11990111989022 minus 1) lt 11990111989011 11990111989022 (3)
That is 119886119887 = 0 (mod11990111989011 11990111989022 )Lemma 4 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 be a positive integer where1199011 1199012 119901119896 are 119896 different primes and 1198901 1198902 119890119896 are 119896positive integers If 119886 = 119901119890119894119894 and 119887 = 119901119890119895119895 with 1 le 119894 119895 le 119896and 119894 = 119895 then 119886119887 = 0 (mod119871)Proof Since 119886 = 119901119890119894119894 and 119887 = 119901119890119895119895 with 1 le 119894 119895 le 119896 and 119894 = 119895then
1 le 119886 times 119887 le 119901119890119894119894 119901119890119895119895 lt 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 = 119871 (4)
That is 119886119887 = 0 (mod119871)Theorem 5 Let 119871 be a positive integer If 119886 and 119887 are twopositive integers and 119886119887 lt 119871 then 119886119887 = 0 (mod119871)Proof Since 119886 and 119887 are two positive integers and 119886119887 lt 119871then
1 le 119886119887 lt 119871 (5)
That is 119886119887 = 0 (mod119871)22 QC LDPC Codes and Their Associated Tanner Graphs A(120574 120588)-regular quasi-cyclic (QC) LDPC code [29] of length 120588119871can be completely specified by the null space of the followingmatrix over GF(2)
H =[[[[[[[[
I (11990100) I (11990101) sdot sdot sdot I (1199010120588minus1)I (11990110) I (11990111) sdot sdot sdot I (1199011120588minus1)
d
I (119901120574minus10) I (119901120574minus11) sdot sdot sdot I (119901120574minus1120588minus1)
]]]]]]]]
(6)
where for 0 le 119894 le 120574 minus 1 and 0 le 119895 le 120588 minus 1 I(119901119894119895) is an 119871 times 119871circulant permutation matrix (CPM) with a one at column-(119903+119901119894119895) (mod119871) for row-119903 0 le 119903 le 119871minus1 and zero elsewhere
Wireless Communications and Mobile Computing 3
H =
[[[[[[[
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
]]]]]]]
(a)
Checknodes
6-cycle
Variablenodes
0
0
1
1
2
2
3
3 4 5(b)
Figure 1 Tanner graph ofH
It is clear that I(0) represents the 119871times119871 identity matrix Noticethat the parameter 119871 is referred to as the expansion factor (orlifting degree) [37] It can be easily observed that the positionsof nonzero elements in H are uniquely determined by thefollowing matrix called permutation shift matrix or exponentmatrix
That is there is a one-to-one correspondence between P andH
An LDPC code is commonly described by a bipartitegraph known as Tanner graph [38] in coding theory Tannergraph ofH denoted byG(119881 119862) consists of a set119881 of variablenodes (containing 120588119871 code symbols of a code word) and a set119862 of check nodes (containing 120574119871 local check-sum constraintson the code symbols) An edge in G(119881 119862) connects thevariable node 119894 to the check node 119895 if and only if the elementat column-119894 and row-119895 of H is nonzero A cycle is formedby a sequence of vertices (or edges) in G(119881 119862) which startsand ends at the same vertex (or edge) and contains othervertices (or edges) not more than once The cycle of length119896 is denoted as 119896-cycle for short and the length of the shortestcycle is called the girth of G(119881 119862) (or an LDPC code) Asan example Figure 1 shows the Tanner graph of H and anassociate 6-cycle
In graph theory the biadjacency matrix A = [119886119894119895] of abipartite graph G(119880 119881) can be constructed as follows Therows ofA are labeled by the |119880| vertices in119880 and the columnsare labeled by |119881| vertices in 119881 The element 119886119894119895 at the rowlabeled by the vertex 119894 isin 119880 and the column labeled by thevertex 119895 isin 119881 is 1 if and only if there exists an edge between thevertices 119894 and 119895 and otherwise 0 Actually for an LDPC codegiven by the null space of H H is the biadjacency matrix ofits relevant Tanner graphG(119881 119862)
Moreover isomorphism theory of QC LDPC codes wasproposed in [39ndash41] based on the isomorphism of graphs ingraph theory According to the isomorphism of QC LDPC
codes the parity-check matrix in (6) can be simplified as thefollowing matrix
H =[[[[[[[[
I (0) I (0) sdot sdot sdot I (0)I (0) I (11990111) sdot sdot sdot I (1199011120588minus1) d
I (0) I (119901120574minus11) sdot sdot sdot I (119901120574minus1120588minus1)
]]]]]]]]
(8)
That is 1199011198940 = 1199010119895 = 0 for 0 le 119894 le 120574 minus 1 0 le 119895 le 120588 minus 1Equivalently its exponent matrix is
That is why the elements in the first row and first columnof the exponent matrix P are usually set to 0 in the researchprocess [29 42] Hence we only consider suchH and P in thefollowing discussions
23 Cycle Structure ofQCLDPCCodes Consider aQCLDPCcodeC given by the null space ofH in (8) It can be seen from[29] that a cycle in the Tanner graph ofC is associated with afamily of the ordered CPMs inH As shown in [29] a 2119894-cyclein the Tanner graph of the codeC (orH) is represented by anordered sequence of CPMs
I (1199011198950 1198960) I (1199011198951 1198960) I (1199011198951 1198961) I (1199011198952 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) I (1199011198950 119896119894minus1) I (1199011198950 1198960)
(10)
where 119895119894 = 1198950 119896119894 = 1198960 0 le 119895119898 le 120574minus 1 119895119898minus1 = 119895119898 0 le 119896119898 le120588 minus 1 and 119896119898minus1 = 119896119898 for 1 le 119898 le 119894 The above sequence canbe simplified as
I (1199011198950 1198960) I (1199011198951 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) (11)
It can be seen that such a 2119894-cycle corresponds to the elements1199011198950 1198960 1199011198951 1198961 1199011198952 1198962 119901119895119894minus1 119896119894minus1 in the exponent matrix P Fur-thermore short cycles of QC LDPC codes can be determinedby the elements of P [39 40]
4 Wireless Communications and Mobile Computing
(pjk) (pj0k2minus1
)
(pj+1k+1)
(pj2minus1k2minus1)
(pj+1k)
pj0k0
pj1k0pj1k1
pj0kminus1
pjminus1kminus1
Figure 2The structure of 2119894-cycle and nonexistence of the 4119894-cycle
Let 119892 be the girth of the codeC It can be seen from [43]that for119892 le 2119894 le 2119892minus2 the necessary and sufficient conditionfor the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) can be generalized as follows
119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = 0 (mod 119871) (12)
with 1198950 = 119895119894 1198960 = 119896119894 119896119898 = 119896119898+1 and 119895119898 = 119895119898+1 Note that(12) is not the sufficient condition for the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) for 2119894 ge 2119892 butit is the necessary condition Here we give a counterexampleConsider a 2119894-cycle (119894 ge 119892) whose cycle structure is given inFigure 2 Clearly (12) is satisfied Let 119901119895119898 119896119898 = 119901119895119898+119894119896119898+119894 for0 le 119898 le 119894 minus 1 and 1198950 = 1198952119894 1198960 = 1198962119894 119896119898 = 119896119898+1 119895119898 = 119895119898+1According to (12) we have
2119894minus1
sum119899=0
(119901119895119899 119896119899 minus 119901119895119899+1 119896119899) = 2119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1119896119898)
= 0 (mod119871) (13)
where 1198950 = 119895119894 = 1198952119894 1198960 = 119896119894 = 1198962119894 119896119899 = 119896119899+1 119895119899 =119895119899+1 for 0 le 119899 le 2119894 minus 1 That is the ordered elements1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 1198962119894minus1 (ie 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1) make (12) hold butthey do not determine a 4119894-cycle A visual representation isdepicted in Figure 2 Therefore (4) in [29] and (3) in [39] arenot applicable to the cycles with lengths larger than 2119892 minus 2
3 Construction of Quasi-Cyclic LDPC Codeswith Girth of at Least 6
Based on the aforementioned there exists a one-to-one cor-respondence between the exponent matrix P and the parity-check matrix H of a QC LDPC code Hence construction ofa QC LDPC code is equivalent to the design of its exponentmatrix P In this section we present three classes of QCLDPC codes with girth of at least 6 and then give a generalconstruction of QC LDPC codes with girth of at least 6 basedon an arbitrary integer
First we design the exponent matrix P in (9) as follows
where 119901119894119895 = 119894 times 119895 (mod119871) for 1 le 119894 le 120574 minus 1 1 le 119895 le 120588 minus 1Second we replace the 0s and 119901119894119895 in the designed exponentmatrix P with CPMs I(0) and I(119901119894119895) of the same size 119871 times 119871respectively and then obtain a 120574 times 120588 arrayH of 119871 times 119871 CPMsThis array is a 120574119871 times 120588119871 matrix over GF(2) with column androwweights 120574 and120588 respectivelyThenull space of thismatrixgives a (120574 120588)-regular QC LDPC code
Remark 6 As shown in [44] girth and short cycles play animportant role in the design of LDPC codes If the aboveconstructed (120574 120588)-regular QC LDPC code does not havegood iterative performance we can replace someCPMs in theabove array H with zero matrices (ZMs) of the same size toreduce the number of short cycles and possibly enlarge thegirth value This replacement is called masking On the otherhand if the lengths of the desiredQCLDPC codes are shorterthan 120588119871 or they require much higher code rates then we cantake a 1205741015840 times 1205881015840 subarray of the designed arrayH where 1205741015840 le 120574and 1205881015840 le 120588 Notice that this subarray can be obtained fromthe following two steps (1) Choose the first 1205741015840 row-CPMsof the designed array H (2) select 1205881015840 column-CPMs from 120588column-CPMs of the designed array H In this paper boththe masking technique and the selection method in [43] areemployed to construct (or further optimize)QCLDPCcodes
31 Three Classes of QC LDPC Codes with Girth of at Least 6Based on (12) we can see that Tanner graph of the designedarrayH contains a 4-cycle if and only if the following equationis satisfied
1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = (1198950 minus 1198951) (1198960 minus 1198961)= 0 (mod119871)
(15)
where 1198950 = 1198951 and 1198960 = 1198961 It can be observed that theexistence of 4-cycles in theTanner graph of the designed arrayH is related to 119871 According to the fundamental theorem ofarithmetic the values of 119871 can be divided into three categoriesand three classes of QC LDPC codes with girth of at least6 are proposed Notice that all numerical simulations in thefollowing examples binary phase shift keying (BPSK) additivewhite Gaussian noise (AWGN) channel and the sum-productalgorithm (SPA) are assumed
311 The Case of 119871 = 119901119890 Let 119871 = 119901119890 where 119901 is a primeand 119890 is a positive integer and let 119890 = 1198901 + 1198902 where 1198901 1198902are two positive integers and 1198901 le 1198902 Consider 120574 = 1199011198901 and120588 = 1199011198902 Since 1 le 1198950 1198951 le 1199011198901 minus 1 1 le 1198960 1198961 le 1199011198902 minus1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 1199011198901 minus 1 and1 le 1198960 minus 1198961 le 1199011198902 minus 1 where the calculation is taken modulo1199011198901 and modulo 1199011198902 respectively Hence (15) is not satisfied
Wireless Communications and Mobile Computing 5
according to Lemma 2 That is Tanner graph of the designedarray H has no 4-cycles and then the constructed QC LDPCcodes have girth of at least 6
Example 7 Consider 119871 = 256 = 28 Let 120574 = 22 and 120588 = 26According to (14) we can obtain the exponentmatrixP of size4 times 64 By employing the method in [43] we select the first 4rows and the 2nd 16th 19th 35th 50th 55th 62nd and 63rdcolumns of P and construct a 4times8 arrayH of 256times256CPMsby replacing the elements of the selected submatrix with thecorresponding CPMs By using the matrix
to mask H a 1024 times 2048 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (2048 1024) QC LDPC code The bit errorrates (BERs) of this code decoded by the SPA (5 10 20 and 50iterations) are shown in Figure 3 Also shown in Figure 3 is theperformance of the (3 6)-regular (2048 1024) algebraic QCLDPC code constructed based on finite fieldGF(119901119904) [35]Thiscomparable code is constructed from the prime field GF(257)and then the CPM size of its parity-check matrix is 256times256Notice that the exponent matrix and masking matrix of thiscomparable code are
respectively It can be observed that these two codes havesimilar performance when decoded using the SPA withvarious iterations It is well known that algebraic-based LDPCcodes have fast decoding convergence [19 45 46]That is theSPAdecoding of the proposed LDPC code also converges fastas shown in Figure 3 We can see that the performance gapbetween 20 and 50 iterations is less than 015 dB at the BER of10minus6 and the gap is also less than 025 dB at the BER of 10minus7hence this code achieves a fast rate of decoding convergence
312 The Case of 119871 = 11990111989011 11990111989022 Let 119871 = 11990111989011 11990111989022 where 1199011 1199012are two different prime numbers and 1198901 1198902 are two positiveintegers Assume 120574 = min11990111989011 11990111989022 and 120588 = max11990111989011 11990111989022 Without loss of generality 11990111989011 lt 11990111989022 is assumed Since 1 le1198950 1198951 le 11990111989011 minus1 1 le 1198960 1198961 le 11990111989022 minus1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950minus1198951 le 11990111989011 minus1 and 1 le 1198960minus1198961 le 11990111989022 minus1 where the calcula-tion is takenmodulo11990111989011 andmodulo11990111989022 respectively Hence(15) is not satisfied according to Lemma 3 That is Tannergraph of the designed array H does not contain 4-cycles andthe girth of the constructed QC LDPC codes is at least 6
Figure 3 The bit error performance of the proposed (3 6)-regular(2048 1024) QC LDPC code and the comparable (3 6)-regular(2048 1024) algebraic QC LDPC code [19] in Example 7 Thedecoding algorithm is the SPA with 5 10 20 and 50 iterations
Example 8 Consider 119871 = 72 = 23times32 = 4times18 Since 12 lt 18let 120574 = 22 and 120588 = 18 According to (14) we can obtain theexponent matrix P of size 4times18 By employing themethod in[43] we select the first 4 rows and the 1st 2nd 3rd 4th 5th6th 12th 13th 15th 16th 17th and 18th columns of P andconstruct a 4 times 12 array H of 72 times 72 CPMs by replacing theelements of the selected submatrix with the correspondingCPMs By using the matrix
to maskH a 288 times 864matrix with column and row weights3 and 9 respectively is obtained This matrix gives a (3 9)-regular (864 576) QC LDPC code The bitword error rates(BERsWERs) of this code decoded by the SPA with 50iterations are shown in Figure 4 Also shown in Figure 4 isthe performance of the (3 9)-regular (864 576) algebraic QCLDPC code constructed from the finite field GF(73) [35]Theexponent and masking matrices of this algebraic QC LDPCcode are
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
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Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 3
H =
[[[[[[[
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1
]]]]]]]
(a)
Checknodes
6-cycle
Variablenodes
0
0
1
1
2
2
3
3 4 5(b)
Figure 1 Tanner graph ofH
It is clear that I(0) represents the 119871times119871 identity matrix Noticethat the parameter 119871 is referred to as the expansion factor (orlifting degree) [37] It can be easily observed that the positionsof nonzero elements in H are uniquely determined by thefollowing matrix called permutation shift matrix or exponentmatrix
That is there is a one-to-one correspondence between P andH
An LDPC code is commonly described by a bipartitegraph known as Tanner graph [38] in coding theory Tannergraph ofH denoted byG(119881 119862) consists of a set119881 of variablenodes (containing 120588119871 code symbols of a code word) and a set119862 of check nodes (containing 120574119871 local check-sum constraintson the code symbols) An edge in G(119881 119862) connects thevariable node 119894 to the check node 119895 if and only if the elementat column-119894 and row-119895 of H is nonzero A cycle is formedby a sequence of vertices (or edges) in G(119881 119862) which startsand ends at the same vertex (or edge) and contains othervertices (or edges) not more than once The cycle of length119896 is denoted as 119896-cycle for short and the length of the shortestcycle is called the girth of G(119881 119862) (or an LDPC code) Asan example Figure 1 shows the Tanner graph of H and anassociate 6-cycle
In graph theory the biadjacency matrix A = [119886119894119895] of abipartite graph G(119880 119881) can be constructed as follows Therows ofA are labeled by the |119880| vertices in119880 and the columnsare labeled by |119881| vertices in 119881 The element 119886119894119895 at the rowlabeled by the vertex 119894 isin 119880 and the column labeled by thevertex 119895 isin 119881 is 1 if and only if there exists an edge between thevertices 119894 and 119895 and otherwise 0 Actually for an LDPC codegiven by the null space of H H is the biadjacency matrix ofits relevant Tanner graphG(119881 119862)
Moreover isomorphism theory of QC LDPC codes wasproposed in [39ndash41] based on the isomorphism of graphs ingraph theory According to the isomorphism of QC LDPC
codes the parity-check matrix in (6) can be simplified as thefollowing matrix
H =[[[[[[[[
I (0) I (0) sdot sdot sdot I (0)I (0) I (11990111) sdot sdot sdot I (1199011120588minus1) d
I (0) I (119901120574minus11) sdot sdot sdot I (119901120574minus1120588minus1)
]]]]]]]]
(8)
That is 1199011198940 = 1199010119895 = 0 for 0 le 119894 le 120574 minus 1 0 le 119895 le 120588 minus 1Equivalently its exponent matrix is
That is why the elements in the first row and first columnof the exponent matrix P are usually set to 0 in the researchprocess [29 42] Hence we only consider suchH and P in thefollowing discussions
23 Cycle Structure ofQCLDPCCodes Consider aQCLDPCcodeC given by the null space ofH in (8) It can be seen from[29] that a cycle in the Tanner graph ofC is associated with afamily of the ordered CPMs inH As shown in [29] a 2119894-cyclein the Tanner graph of the codeC (orH) is represented by anordered sequence of CPMs
I (1199011198950 1198960) I (1199011198951 1198960) I (1199011198951 1198961) I (1199011198952 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) I (1199011198950 119896119894minus1) I (1199011198950 1198960)
(10)
where 119895119894 = 1198950 119896119894 = 1198960 0 le 119895119898 le 120574minus 1 119895119898minus1 = 119895119898 0 le 119896119898 le120588 minus 1 and 119896119898minus1 = 119896119898 for 1 le 119898 le 119894 The above sequence canbe simplified as
I (1199011198950 1198960) I (1199011198951 1198961) I (1199011198952 1198962) I (119901119895119894minus1119896119894minus1) (11)
It can be seen that such a 2119894-cycle corresponds to the elements1199011198950 1198960 1199011198951 1198961 1199011198952 1198962 119901119895119894minus1 119896119894minus1 in the exponent matrix P Fur-thermore short cycles of QC LDPC codes can be determinedby the elements of P [39 40]
4 Wireless Communications and Mobile Computing
(pjk) (pj0k2minus1
)
(pj+1k+1)
(pj2minus1k2minus1)
(pj+1k)
pj0k0
pj1k0pj1k1
pj0kminus1
pjminus1kminus1
Figure 2The structure of 2119894-cycle and nonexistence of the 4119894-cycle
Let 119892 be the girth of the codeC It can be seen from [43]that for119892 le 2119894 le 2119892minus2 the necessary and sufficient conditionfor the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) can be generalized as follows
119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = 0 (mod 119871) (12)
with 1198950 = 119895119894 1198960 = 119896119894 119896119898 = 119896119898+1 and 119895119898 = 119895119898+1 Note that(12) is not the sufficient condition for the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) for 2119894 ge 2119892 butit is the necessary condition Here we give a counterexampleConsider a 2119894-cycle (119894 ge 119892) whose cycle structure is given inFigure 2 Clearly (12) is satisfied Let 119901119895119898 119896119898 = 119901119895119898+119894119896119898+119894 for0 le 119898 le 119894 minus 1 and 1198950 = 1198952119894 1198960 = 1198962119894 119896119898 = 119896119898+1 119895119898 = 119895119898+1According to (12) we have
2119894minus1
sum119899=0
(119901119895119899 119896119899 minus 119901119895119899+1 119896119899) = 2119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1119896119898)
= 0 (mod119871) (13)
where 1198950 = 119895119894 = 1198952119894 1198960 = 119896119894 = 1198962119894 119896119899 = 119896119899+1 119895119899 =119895119899+1 for 0 le 119899 le 2119894 minus 1 That is the ordered elements1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 1198962119894minus1 (ie 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1) make (12) hold butthey do not determine a 4119894-cycle A visual representation isdepicted in Figure 2 Therefore (4) in [29] and (3) in [39] arenot applicable to the cycles with lengths larger than 2119892 minus 2
3 Construction of Quasi-Cyclic LDPC Codeswith Girth of at Least 6
Based on the aforementioned there exists a one-to-one cor-respondence between the exponent matrix P and the parity-check matrix H of a QC LDPC code Hence construction ofa QC LDPC code is equivalent to the design of its exponentmatrix P In this section we present three classes of QCLDPC codes with girth of at least 6 and then give a generalconstruction of QC LDPC codes with girth of at least 6 basedon an arbitrary integer
First we design the exponent matrix P in (9) as follows
where 119901119894119895 = 119894 times 119895 (mod119871) for 1 le 119894 le 120574 minus 1 1 le 119895 le 120588 minus 1Second we replace the 0s and 119901119894119895 in the designed exponentmatrix P with CPMs I(0) and I(119901119894119895) of the same size 119871 times 119871respectively and then obtain a 120574 times 120588 arrayH of 119871 times 119871 CPMsThis array is a 120574119871 times 120588119871 matrix over GF(2) with column androwweights 120574 and120588 respectivelyThenull space of thismatrixgives a (120574 120588)-regular QC LDPC code
Remark 6 As shown in [44] girth and short cycles play animportant role in the design of LDPC codes If the aboveconstructed (120574 120588)-regular QC LDPC code does not havegood iterative performance we can replace someCPMs in theabove array H with zero matrices (ZMs) of the same size toreduce the number of short cycles and possibly enlarge thegirth value This replacement is called masking On the otherhand if the lengths of the desiredQCLDPC codes are shorterthan 120588119871 or they require much higher code rates then we cantake a 1205741015840 times 1205881015840 subarray of the designed arrayH where 1205741015840 le 120574and 1205881015840 le 120588 Notice that this subarray can be obtained fromthe following two steps (1) Choose the first 1205741015840 row-CPMsof the designed array H (2) select 1205881015840 column-CPMs from 120588column-CPMs of the designed array H In this paper boththe masking technique and the selection method in [43] areemployed to construct (or further optimize)QCLDPCcodes
31 Three Classes of QC LDPC Codes with Girth of at Least 6Based on (12) we can see that Tanner graph of the designedarrayH contains a 4-cycle if and only if the following equationis satisfied
1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = (1198950 minus 1198951) (1198960 minus 1198961)= 0 (mod119871)
(15)
where 1198950 = 1198951 and 1198960 = 1198961 It can be observed that theexistence of 4-cycles in theTanner graph of the designed arrayH is related to 119871 According to the fundamental theorem ofarithmetic the values of 119871 can be divided into three categoriesand three classes of QC LDPC codes with girth of at least6 are proposed Notice that all numerical simulations in thefollowing examples binary phase shift keying (BPSK) additivewhite Gaussian noise (AWGN) channel and the sum-productalgorithm (SPA) are assumed
311 The Case of 119871 = 119901119890 Let 119871 = 119901119890 where 119901 is a primeand 119890 is a positive integer and let 119890 = 1198901 + 1198902 where 1198901 1198902are two positive integers and 1198901 le 1198902 Consider 120574 = 1199011198901 and120588 = 1199011198902 Since 1 le 1198950 1198951 le 1199011198901 minus 1 1 le 1198960 1198961 le 1199011198902 minus1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 1199011198901 minus 1 and1 le 1198960 minus 1198961 le 1199011198902 minus 1 where the calculation is taken modulo1199011198901 and modulo 1199011198902 respectively Hence (15) is not satisfied
Wireless Communications and Mobile Computing 5
according to Lemma 2 That is Tanner graph of the designedarray H has no 4-cycles and then the constructed QC LDPCcodes have girth of at least 6
Example 7 Consider 119871 = 256 = 28 Let 120574 = 22 and 120588 = 26According to (14) we can obtain the exponentmatrixP of size4 times 64 By employing the method in [43] we select the first 4rows and the 2nd 16th 19th 35th 50th 55th 62nd and 63rdcolumns of P and construct a 4times8 arrayH of 256times256CPMsby replacing the elements of the selected submatrix with thecorresponding CPMs By using the matrix
to mask H a 1024 times 2048 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (2048 1024) QC LDPC code The bit errorrates (BERs) of this code decoded by the SPA (5 10 20 and 50iterations) are shown in Figure 3 Also shown in Figure 3 is theperformance of the (3 6)-regular (2048 1024) algebraic QCLDPC code constructed based on finite fieldGF(119901119904) [35]Thiscomparable code is constructed from the prime field GF(257)and then the CPM size of its parity-check matrix is 256times256Notice that the exponent matrix and masking matrix of thiscomparable code are
respectively It can be observed that these two codes havesimilar performance when decoded using the SPA withvarious iterations It is well known that algebraic-based LDPCcodes have fast decoding convergence [19 45 46]That is theSPAdecoding of the proposed LDPC code also converges fastas shown in Figure 3 We can see that the performance gapbetween 20 and 50 iterations is less than 015 dB at the BER of10minus6 and the gap is also less than 025 dB at the BER of 10minus7hence this code achieves a fast rate of decoding convergence
312 The Case of 119871 = 11990111989011 11990111989022 Let 119871 = 11990111989011 11990111989022 where 1199011 1199012are two different prime numbers and 1198901 1198902 are two positiveintegers Assume 120574 = min11990111989011 11990111989022 and 120588 = max11990111989011 11990111989022 Without loss of generality 11990111989011 lt 11990111989022 is assumed Since 1 le1198950 1198951 le 11990111989011 minus1 1 le 1198960 1198961 le 11990111989022 minus1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950minus1198951 le 11990111989011 minus1 and 1 le 1198960minus1198961 le 11990111989022 minus1 where the calcula-tion is takenmodulo11990111989011 andmodulo11990111989022 respectively Hence(15) is not satisfied according to Lemma 3 That is Tannergraph of the designed array H does not contain 4-cycles andthe girth of the constructed QC LDPC codes is at least 6
Figure 3 The bit error performance of the proposed (3 6)-regular(2048 1024) QC LDPC code and the comparable (3 6)-regular(2048 1024) algebraic QC LDPC code [19] in Example 7 Thedecoding algorithm is the SPA with 5 10 20 and 50 iterations
Example 8 Consider 119871 = 72 = 23times32 = 4times18 Since 12 lt 18let 120574 = 22 and 120588 = 18 According to (14) we can obtain theexponent matrix P of size 4times18 By employing themethod in[43] we select the first 4 rows and the 1st 2nd 3rd 4th 5th6th 12th 13th 15th 16th 17th and 18th columns of P andconstruct a 4 times 12 array H of 72 times 72 CPMs by replacing theelements of the selected submatrix with the correspondingCPMs By using the matrix
to maskH a 288 times 864matrix with column and row weights3 and 9 respectively is obtained This matrix gives a (3 9)-regular (864 576) QC LDPC code The bitword error rates(BERsWERs) of this code decoded by the SPA with 50iterations are shown in Figure 4 Also shown in Figure 4 isthe performance of the (3 9)-regular (864 576) algebraic QCLDPC code constructed from the finite field GF(73) [35]Theexponent and masking matrices of this algebraic QC LDPCcode are
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
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Submit your manuscripts atwwwhindawicom
4 Wireless Communications and Mobile Computing
(pjk) (pj0k2minus1
)
(pj+1k+1)
(pj2minus1k2minus1)
(pj+1k)
pj0k0
pj1k0pj1k1
pj0kminus1
pjminus1kminus1
Figure 2The structure of 2119894-cycle and nonexistence of the 4119894-cycle
Let 119892 be the girth of the codeC It can be seen from [43]that for119892 le 2119894 le 2119892minus2 the necessary and sufficient conditionfor the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) can be generalized as follows
119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = 0 (mod 119871) (12)
with 1198950 = 119895119894 1198960 = 119896119894 119896119898 = 119896119898+1 and 119895119898 = 119895119898+1 Note that(12) is not the sufficient condition for the existence of a 2119894-cycle in the Tanner graph of the codeC (orH) for 2119894 ge 2119892 butit is the necessary condition Here we give a counterexampleConsider a 2119894-cycle (119894 ge 119892) whose cycle structure is given inFigure 2 Clearly (12) is satisfied Let 119901119895119898 119896119898 = 119901119895119898+119894119896119898+119894 for0 le 119898 le 119894 minus 1 and 1198950 = 1198952119894 1198960 = 1198962119894 119896119898 = 119896119898+1 119895119898 = 119895119898+1According to (12) we have
2119894minus1
sum119899=0
(119901119895119899 119896119899 minus 119901119895119899+1 119896119899) = 2119894minus1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1119896119898)
= 0 (mod119871) (13)
where 1198950 = 119895119894 = 1198952119894 1198960 = 119896119894 = 1198962119894 119896119899 = 119896119899+1 119895119899 =119895119899+1 for 0 le 119899 le 2119894 minus 1 That is the ordered elements1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 1198962119894minus1 (ie 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1 1199011198950 1198960 1199011198951 1198960 1199011198951 1198961 1199011198950 119896119894minus1) make (12) hold butthey do not determine a 4119894-cycle A visual representation isdepicted in Figure 2 Therefore (4) in [29] and (3) in [39] arenot applicable to the cycles with lengths larger than 2119892 minus 2
3 Construction of Quasi-Cyclic LDPC Codeswith Girth of at Least 6
Based on the aforementioned there exists a one-to-one cor-respondence between the exponent matrix P and the parity-check matrix H of a QC LDPC code Hence construction ofa QC LDPC code is equivalent to the design of its exponentmatrix P In this section we present three classes of QCLDPC codes with girth of at least 6 and then give a generalconstruction of QC LDPC codes with girth of at least 6 basedon an arbitrary integer
First we design the exponent matrix P in (9) as follows
where 119901119894119895 = 119894 times 119895 (mod119871) for 1 le 119894 le 120574 minus 1 1 le 119895 le 120588 minus 1Second we replace the 0s and 119901119894119895 in the designed exponentmatrix P with CPMs I(0) and I(119901119894119895) of the same size 119871 times 119871respectively and then obtain a 120574 times 120588 arrayH of 119871 times 119871 CPMsThis array is a 120574119871 times 120588119871 matrix over GF(2) with column androwweights 120574 and120588 respectivelyThenull space of thismatrixgives a (120574 120588)-regular QC LDPC code
Remark 6 As shown in [44] girth and short cycles play animportant role in the design of LDPC codes If the aboveconstructed (120574 120588)-regular QC LDPC code does not havegood iterative performance we can replace someCPMs in theabove array H with zero matrices (ZMs) of the same size toreduce the number of short cycles and possibly enlarge thegirth value This replacement is called masking On the otherhand if the lengths of the desiredQCLDPC codes are shorterthan 120588119871 or they require much higher code rates then we cantake a 1205741015840 times 1205881015840 subarray of the designed arrayH where 1205741015840 le 120574and 1205881015840 le 120588 Notice that this subarray can be obtained fromthe following two steps (1) Choose the first 1205741015840 row-CPMsof the designed array H (2) select 1205881015840 column-CPMs from 120588column-CPMs of the designed array H In this paper boththe masking technique and the selection method in [43] areemployed to construct (or further optimize)QCLDPCcodes
31 Three Classes of QC LDPC Codes with Girth of at Least 6Based on (12) we can see that Tanner graph of the designedarrayH contains a 4-cycle if and only if the following equationis satisfied
1
sum119898=0
(119901119895119898 119896119898 minus 119901119895119898+1 119896119898) = (1198950 minus 1198951) (1198960 minus 1198961)= 0 (mod119871)
(15)
where 1198950 = 1198951 and 1198960 = 1198961 It can be observed that theexistence of 4-cycles in theTanner graph of the designed arrayH is related to 119871 According to the fundamental theorem ofarithmetic the values of 119871 can be divided into three categoriesand three classes of QC LDPC codes with girth of at least6 are proposed Notice that all numerical simulations in thefollowing examples binary phase shift keying (BPSK) additivewhite Gaussian noise (AWGN) channel and the sum-productalgorithm (SPA) are assumed
311 The Case of 119871 = 119901119890 Let 119871 = 119901119890 where 119901 is a primeand 119890 is a positive integer and let 119890 = 1198901 + 1198902 where 1198901 1198902are two positive integers and 1198901 le 1198902 Consider 120574 = 1199011198901 and120588 = 1199011198902 Since 1 le 1198950 1198951 le 1199011198901 minus 1 1 le 1198960 1198961 le 1199011198902 minus1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 1199011198901 minus 1 and1 le 1198960 minus 1198961 le 1199011198902 minus 1 where the calculation is taken modulo1199011198901 and modulo 1199011198902 respectively Hence (15) is not satisfied
Wireless Communications and Mobile Computing 5
according to Lemma 2 That is Tanner graph of the designedarray H has no 4-cycles and then the constructed QC LDPCcodes have girth of at least 6
Example 7 Consider 119871 = 256 = 28 Let 120574 = 22 and 120588 = 26According to (14) we can obtain the exponentmatrixP of size4 times 64 By employing the method in [43] we select the first 4rows and the 2nd 16th 19th 35th 50th 55th 62nd and 63rdcolumns of P and construct a 4times8 arrayH of 256times256CPMsby replacing the elements of the selected submatrix with thecorresponding CPMs By using the matrix
to mask H a 1024 times 2048 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (2048 1024) QC LDPC code The bit errorrates (BERs) of this code decoded by the SPA (5 10 20 and 50iterations) are shown in Figure 3 Also shown in Figure 3 is theperformance of the (3 6)-regular (2048 1024) algebraic QCLDPC code constructed based on finite fieldGF(119901119904) [35]Thiscomparable code is constructed from the prime field GF(257)and then the CPM size of its parity-check matrix is 256times256Notice that the exponent matrix and masking matrix of thiscomparable code are
respectively It can be observed that these two codes havesimilar performance when decoded using the SPA withvarious iterations It is well known that algebraic-based LDPCcodes have fast decoding convergence [19 45 46]That is theSPAdecoding of the proposed LDPC code also converges fastas shown in Figure 3 We can see that the performance gapbetween 20 and 50 iterations is less than 015 dB at the BER of10minus6 and the gap is also less than 025 dB at the BER of 10minus7hence this code achieves a fast rate of decoding convergence
312 The Case of 119871 = 11990111989011 11990111989022 Let 119871 = 11990111989011 11990111989022 where 1199011 1199012are two different prime numbers and 1198901 1198902 are two positiveintegers Assume 120574 = min11990111989011 11990111989022 and 120588 = max11990111989011 11990111989022 Without loss of generality 11990111989011 lt 11990111989022 is assumed Since 1 le1198950 1198951 le 11990111989011 minus1 1 le 1198960 1198961 le 11990111989022 minus1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950minus1198951 le 11990111989011 minus1 and 1 le 1198960minus1198961 le 11990111989022 minus1 where the calcula-tion is takenmodulo11990111989011 andmodulo11990111989022 respectively Hence(15) is not satisfied according to Lemma 3 That is Tannergraph of the designed array H does not contain 4-cycles andthe girth of the constructed QC LDPC codes is at least 6
Figure 3 The bit error performance of the proposed (3 6)-regular(2048 1024) QC LDPC code and the comparable (3 6)-regular(2048 1024) algebraic QC LDPC code [19] in Example 7 Thedecoding algorithm is the SPA with 5 10 20 and 50 iterations
Example 8 Consider 119871 = 72 = 23times32 = 4times18 Since 12 lt 18let 120574 = 22 and 120588 = 18 According to (14) we can obtain theexponent matrix P of size 4times18 By employing themethod in[43] we select the first 4 rows and the 1st 2nd 3rd 4th 5th6th 12th 13th 15th 16th 17th and 18th columns of P andconstruct a 4 times 12 array H of 72 times 72 CPMs by replacing theelements of the selected submatrix with the correspondingCPMs By using the matrix
to maskH a 288 times 864matrix with column and row weights3 and 9 respectively is obtained This matrix gives a (3 9)-regular (864 576) QC LDPC code The bitword error rates(BERsWERs) of this code decoded by the SPA with 50iterations are shown in Figure 4 Also shown in Figure 4 isthe performance of the (3 9)-regular (864 576) algebraic QCLDPC code constructed from the finite field GF(73) [35]Theexponent and masking matrices of this algebraic QC LDPCcode are
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
Propagation
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Navigation and Observation
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wwwhindawicom Volume 2018
Advances in
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Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 5
according to Lemma 2 That is Tanner graph of the designedarray H has no 4-cycles and then the constructed QC LDPCcodes have girth of at least 6
Example 7 Consider 119871 = 256 = 28 Let 120574 = 22 and 120588 = 26According to (14) we can obtain the exponentmatrixP of size4 times 64 By employing the method in [43] we select the first 4rows and the 2nd 16th 19th 35th 50th 55th 62nd and 63rdcolumns of P and construct a 4times8 arrayH of 256times256CPMsby replacing the elements of the selected submatrix with thecorresponding CPMs By using the matrix
to mask H a 1024 times 2048 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (2048 1024) QC LDPC code The bit errorrates (BERs) of this code decoded by the SPA (5 10 20 and 50iterations) are shown in Figure 3 Also shown in Figure 3 is theperformance of the (3 6)-regular (2048 1024) algebraic QCLDPC code constructed based on finite fieldGF(119901119904) [35]Thiscomparable code is constructed from the prime field GF(257)and then the CPM size of its parity-check matrix is 256times256Notice that the exponent matrix and masking matrix of thiscomparable code are
respectively It can be observed that these two codes havesimilar performance when decoded using the SPA withvarious iterations It is well known that algebraic-based LDPCcodes have fast decoding convergence [19 45 46]That is theSPAdecoding of the proposed LDPC code also converges fastas shown in Figure 3 We can see that the performance gapbetween 20 and 50 iterations is less than 015 dB at the BER of10minus6 and the gap is also less than 025 dB at the BER of 10minus7hence this code achieves a fast rate of decoding convergence
312 The Case of 119871 = 11990111989011 11990111989022 Let 119871 = 11990111989011 11990111989022 where 1199011 1199012are two different prime numbers and 1198901 1198902 are two positiveintegers Assume 120574 = min11990111989011 11990111989022 and 120588 = max11990111989011 11990111989022 Without loss of generality 11990111989011 lt 11990111989022 is assumed Since 1 le1198950 1198951 le 11990111989011 minus1 1 le 1198960 1198961 le 11990111989022 minus1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950minus1198951 le 11990111989011 minus1 and 1 le 1198960minus1198961 le 11990111989022 minus1 where the calcula-tion is takenmodulo11990111989011 andmodulo11990111989022 respectively Hence(15) is not satisfied according to Lemma 3 That is Tannergraph of the designed array H does not contain 4-cycles andthe girth of the constructed QC LDPC codes is at least 6
Figure 3 The bit error performance of the proposed (3 6)-regular(2048 1024) QC LDPC code and the comparable (3 6)-regular(2048 1024) algebraic QC LDPC code [19] in Example 7 Thedecoding algorithm is the SPA with 5 10 20 and 50 iterations
Example 8 Consider 119871 = 72 = 23times32 = 4times18 Since 12 lt 18let 120574 = 22 and 120588 = 18 According to (14) we can obtain theexponent matrix P of size 4times18 By employing themethod in[43] we select the first 4 rows and the 1st 2nd 3rd 4th 5th6th 12th 13th 15th 16th 17th and 18th columns of P andconstruct a 4 times 12 array H of 72 times 72 CPMs by replacing theelements of the selected submatrix with the correspondingCPMs By using the matrix
to maskH a 288 times 864matrix with column and row weights3 and 9 respectively is obtained This matrix gives a (3 9)-regular (864 576) QC LDPC code The bitword error rates(BERsWERs) of this code decoded by the SPA with 50iterations are shown in Figure 4 Also shown in Figure 4 isthe performance of the (3 9)-regular (864 576) algebraic QCLDPC code constructed from the finite field GF(73) [35]Theexponent and masking matrices of this algebraic QC LDPCcode are
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Figure 4 The error performance of the proposed (3 9)-regular(864 576) QC LDPC code and the comparable (3 9)-regular(864 576) algebraic QC LDPC code constructed based on finite fieldGF (73) [35] in Example 8
respectively Notice that the CPM size of this algebraic codeis 72 times 72 It can be observed that these two codes alsohave similar performance Moreover the cycle distributionsof these two codes are given in Table 1 We can see thatalthough the proposed code has fewer shortest cycles thanthe algebraic QC LDPC code the proposed code has muchmore cycles of length 8 than the algebraic QC LDPC codeThat is why the proposed code does not perform better thanthe algebraic QC LDPC code in the high-SNR region
313 The Case of 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 Let 119871 = 11990111989011 11990111989022 sdot sdot sdot 119901119890119896119896 where 1199011 1199012 119901119896 are 119896 different prime numbers and1198901 1198902 119890119896 are 119896 positive integersWithout loss of generalitywe assume 119901119890119894119894 lt 119901119890119895119895 where 1 le 119894 119895 le 119896 and 119894 = 119895 Consider120574 = 119901119890119894119894 and 120588 = 119901119890119895119895 Since 1 le 1198950 1198951 le 119901119890119894119894 minus 1 1 le 1198960 1198961 le119901119890119895119895 minus 1 1198950 = 1198951 and 1198960 = 1198961 then 1 le 1198950 minus 1198951 le 119901119890119894119894 minus 1 and1 le 1198960 minus 1198961 le 119901119890119895119895 minus 1 where the calculation is taken modulo119901119890119894119894 and modulo 119901119890119895119895 respectively Hence (15) is not satisfied
according to Lemma 4That is Tanner graph of the designedarrayH does not have 4-cycles and the constructedQCLDPCcodes have girth of at least 6
Example 9 Consider 119871 = 105 = 3 times 5 times 7 Since 10 lt 21let 120574 = 5 and 120588 = 21 According to (14) we can obtain theexponent matrix P of size 5 times 21 By employing the methodin [43] we select the first 5 rows and the 1st 2nd 3rd 6th 7th13th 16th 17th 20th and 21st columns of P and construct a5 times 10 arrayH of 105 times 105 CPMs by replacing the elementsof the selected submatrix with the corresponding CPMs Byusing the matrix
to maskH a 525times1050matrix with column and row weights3 and 6 respectively is obtained This matrix gives a (3 6)-regular (1050 525) QC LDPC code of girth 8 For compari-son we simultaneously present the simulation for the (3 6)-regular (1050 525) LDPC code constructed based on theprogressive edge-growth (PEG) algorithm [22] The bitworderror rates (BERsWERs) of these two codes decoded withthe SPA (50 iterations) are shown in Figure 5 It can be seenthat although these two codes have similar performance inthe waterfall region the proposed code performs better thanthe PEG-LDPC code in the high-SNR region
32 A General Construction of QC LDPC Codes from anArbitrary Positive Integer For a given positive integer 119871 wein general find out two positive integers 119886 and 119887 such that119886119887 le 119871 and 119886 119887 ge 3 Assume 119886 le 119887 Consider 120574 = 119886 and120588 = 119887 where 1 le 119894 119895 le 119896 and 119894 = 119895 Since 1 le 1198950 1198951 le119886 minus 1 1 le 1198960 1198961 le 119887 minus 1 1198950 = 1198951 and 1198960 = 1198961 then1 le 1198950 minus 1198951 le 119886 minus 1 and 1 le 1198960 minus 1198961 le 119887 minus 1 wherethe calculation is takenmodulo 119886 andmodulo 119887 respectivelyHence (15) is not satisfied according to Theorem 5 That isTanner graph of the designed arrayH does not have 4-cyclesand the constructed QC LDPC codes have girth of at least 6
Example 10 Consider 119871 = 127 gt 4 times 31 and let 120574 = 4 120588 =31 According to (14) we can obtain the exponent matrix Pof size 4 times 31 By employing the method in [43] we select thefirst 4 rows and the 1st 2nd 6th 7th 22nd 26th 29th and 31stcolumns of P and construct a 4times8 arrayH of 127times127CPMsby replacing the elements of the selected submatrix with the
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Wireless Communications and Mobile Computing 7BE
RW
ER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER PEGWER PEG
BER proposedWER proposed
Figure 5 The error performance of the proposed (3 6)-regular(1050 525) QC LDPC code and the comparable (3 6)-regular(1050 525)QCLDPCcode constructed based on the PEGalgorithm[22] in Example 9
corresponding CPMs By using themethod in [43] we designa masking matrix that is
to mask H a 508 times 1016 matrix with column and rowweights 3 and 6 respectively is obtained This matrix givesa (3 6)-regular (1016 508) QC LDPC code of girth 8 Forcomparison we also construct a (3 6)-regular (1016 508)QCLDPC code based on the partial geometry [28] Note that theexponent matrix of this code is
and the masking matrix is also M4times8 in Example 7 Thebitword error performance of these two codes decoded bythe SPA with 50 iterations is shown in Figure 6 It can be seenthat these two codes have similar performance We can alsoobserve from Figure 6 that for the proposed QC LDPC codethere are no error floors in the BER curves down to BER =227times10minus7 and in theWER curves down toWER= 35times10minus64 Conclusion
In this paper based on the fundamental theorem of arith-metic we presented a method for constructing QC LDPCcodes with girth of at least 6 from an arbitrary integerAccording to the integer factorization we divided the integers
BER
WER
100
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6
10minus71 125 15 175 2 225 25 275 3
EbN0 (dB)
BER partial geometryWER partial geometry
BER proposedWER proposed
Figure 6 The bit error performance of the proposed (3 6)-regular(1016 508) QC LDPC code and the comparable (3 6)-regular(1016 508)QC LDPC code constructed from partial geometry [28]in Example 10
into three categories and then constructed three classes ofQC LDPC codes Furthermore a general construction of QCLDPC codes with girth of at least 6 was proposed Numericalresults show that the constructed QC LDPC codes have goodperformance over the AWGN channel and converge fastunder iterative decoding In other words for an arbitraryinteger 119871(ge 6) we can easily construct QC LDPC codeswhose parity-check matrices consist of several CPMs andorzero matrices of size 119871times119871 and the proposedmethod ensuredthat the resultant QC LDPC codes have girth of at least 6Moreover the proposed QC LDPC codes perform as well asthe algebraic QC LDPC codes
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NaturalScience Foundation of China under Grant 61103143 the JointFunds of the National Natural Science Foundation of Chinaunder Grant U1504601 the Key Scientific and TechnologicalProject of Henan under Grants 162102310589 172102310124and 182102310867 the Key Scientific Research Projects ofHenan Educational Committee under Grant 18B510022 andthe School-Based Program of Zhoukou Normal Universityunder Grant ZKNUB2201705
References
[1] R G Gallager ldquoLow-Density Parity-Check Codesrdquo IRE Trans-actions on Information Theory vol 8 no 1 pp 21ndash28 1962
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Wireless Communications and Mobile Computing
[2] IEEE Standard ldquoAir Interface for Fixed Broadband WirelessAccess Systemsrdquo IEEE Standard P80216eD1 2005
[3] European Telecommunications Standards Institute DigitalVideo Broadcasting (DVB) European TelecommunicationsStandards Institute Sophia Antipolis France 2009
[4] CCSDS ldquoShort Block Length LDPCCodes for TC Synchroniza-tion and Channel Codingrdquo CCSDS 2311-O-1 2015
[13] M Zhang Z Wang Q Huang and S Wang ldquoTime-InvariantQuasi-Cyclic Spatially Coupled LDPC Codes Based on Pack-ingsrdquo IEEE Transactions on Communications vol 64 no 12 pp4936ndash4945 2016
[14] X Ma K Huang and B Bai ldquoSystematic block Markov super-position transmission of repetition codesrdquo IEEETransactions onInformation Theory vol 64 no 3 pp 1604ndash1620 2018
[15] B Bai ldquoNonbinary LDPC coding for 5G communicationsystemsrdquo in Proceedings of the 10th International Conference onInformation Communications and Signal Processing (ICICSrsquo15)pp 2ndash4 Singapore 2015
[16] S Wang Q Huang and Z Wang ldquoSymbol flipping decodingalgorithms based on prediction for non-binary LDPC codesrdquoIEEE Transactions on Communications vol 65 no 5 pp 1913ndash1924 2017
[17] Q Huang L Song and Z Wang ldquoSet Message-Passing Decod-ing Algorithms for Regular Non-Binary LDPC Codesrdquo IEEETransactions on Communications 2017
[18] 3GPP ldquoStudy on scenarios and requirements for next generationaccess technologiesrdquo Technical Report (TR) 38913 2016
[19] W Ryan and S Lin Channel Codes Classical and ModernCambridge University Press New York NY USA 2009
[20] L Lan Y Y Tai S Lin B Memari and B Honary ldquoNewconstructions of quasi-cyclic LDPC codes based on specialclasses of BIBDrsquos for the AWGN and binary erasure channelsrdquoIEEE Transactions on Communications vol 56 no 1 pp 39ndash482008
[21] T Tian C Jones J D Villasenor and R D Wesel ldquoCon-struction of irregular LDPC codes with low error floorsrdquo inProceedings of the International Conference on Communications(ICCrsquo03) pp 3125ndash3129 2003
[22] X-Y Hu E Eleftheriou and D M Arnold ldquoRegular andirregular progressive edge-growth Tanner graphsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 1 pp 386ndash398 2005
[23] Q Diao Y Y Tai S Lin and K Abdel-Ghaffar ldquoLDPC codeson partial geometries construction trapping set structure and
puncturingrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 59 no 12 pp 7898ndash7914 2013
[24] S Song B Zhou S Lin and K Abdel-Ghaffar ldquoA unifiedapproach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fieldsrdquo IEEE Transactions onCommunications vol 57 no 1 pp 84ndash93 2009
[25] H Xu D Feng C Sun and B Bai ldquoConstruction of LDPCcodes based on resolvable group divisible designsrdquo in Proceed-ings of the International Workshop on High Mobility WirelessCommunications (HMWCrsquo15) pp 111ndash115 2015
[26] D Divsalar S Dolinar C R Jones and K Andrews ldquoCapacity-approaching protograph codesrdquo IEEE Journal on Selected Areasin Communications vol 27 no 6 pp 876ndash888 2009
[27] D G Mitchell R Smarandache and J Costello ldquoQuasi-cyclicLDPC codes based on pre-lifted protographsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 60 no 10 pp 5856ndash5874 2014
[28] Q Diao J Li S Lin and I F Blake ldquoNew classes of partialgeometries and their associated LDPC codesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 62 no 6 pp 2947ndash2965 2016
[29] M P Fossorier ldquoQuasi-cyclic low-density parity-check codesfrom circulant permutation matricesrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol50 no 8 pp 1788ndash1793 2004
[30] Z Li L Chen L Zeng S Lin and W H Fong ldquoEfficientencoding of quasi-cyclic low-density parity-check codesrdquo IEEETransactions on Communications vol 54 no 1 pp 71ndash81 2006
[31] J Li K Liu S Lin and K Abdel-Ghaffar ldquoDecoding ofquasi-cyclic LDPC codes with section-wise cyclic structurerdquo inProceedings of the IEEE Information Theory and ApplicationsWorkshop (ITArsquo14) pp 1ndash10 Calif USA 2014
[32] F Cai X ZhangDDeclercq S K Planjery andBVasic ldquoFinitealphabet iterative decoders for LDPC codes optimizationarchitecture and analysisrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 5 pp 1366ndash1375 2014
[33] H Liu Q Huang G Deng and J Chen ldquoQuasi-cyclic repre-sentation and vector representation of RS-LDPC Codesrdquo IEEETransactions on Communications vol 63 no 4 pp 1033ndash10422015
[34] Q Huang L Tang S He Z Xiong and Z Wang ldquoLow-complexity encoding of quasi-cyclic codes based on GaloisFourier transformrdquo IEEE Transactions on Communications vol62 no 6 pp 1757ndash1767 2014
[35] J Li K Liu S Lin and K Abdel-Ghaffar ldquoAlgebraic quasi-cyclic ldpc codes Construction low error-floor large girth anda reduced-complexity decoding schemerdquo IEEE Transactions onCommunications vol 62 no 8 pp 2626ndash2637 2014
[36] C F Gauss and A A Clarke Disquisitiones arithmeticae(Second corrected edition) springer New York NY USA 1966
[37] J Li K Liu S Lin K Abdel-Ghaffar and W E Ryan ldquoAnunnoticed strong connection between algebraic-based and pro-tograph-based LDPC codes Part I Binary case and interpreta-tionrdquo in Proceedings of the Information Theory and ApplicationsWorkshop (ITArsquo15) pp 36ndash45 San Diego Calif USA 2015
[38] R M Tanner ldquoA recursive approach to low complexity codesrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 27 no 5 pp 533ndash547 1981
[39] A Tasdighi A H Banihashemi and M-R Sadeghi ldquoEfficientsearch of girth-optimal QC-LDPC codesrdquo Institute of Electrical
Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
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Wireless Communications and Mobile Computing 9
and Electronics Engineers Transactions on Information Theoryvol 62 no 4 pp 1552ndash1564 2016
[40] C Sun H Xu D Feng and B Bai ldquo(3 L) quasi-cyclic LDPCcodes Simplified exhaustive search and designsrdquo in Proceedingsof the 9th International Symposium on Turbo Codes and IterativeInformation Processing (ISTCrsquo16) pp 271ndash275 Brest France2016
[41] H Xu C Chen M Zhu B M Bai and B Zhang ldquoNonbinaryLDPC cycle codes Efficient search design and code optimiza-tionrdquo Science China Information Sciences httpenginescichinacomdoi101007s11432-017-9271-6
[42] S Zhao and X Ma ldquoConstruction of high-performance array-based non-binary LDPC codes with moderate ratesrdquo IEEECommunications Letters vol 20 no 1 pp 13ndash16 2016
[43] H Xu D Feng R Luo and B Bai ldquoConstruction of quasi-cyclicLDPC codes via masking with successive cycle eliminationrdquoIEEE Communications Letters vol 20 no 12 pp 2370ndash23732016
[44] H Xu and B Bai ldquoSuperposition Construction of Q-Ary LDPCCodes by Jointly Optimizing Girth and Number of ShortestCyclesrdquo IEEE Communications Letters vol 20 no 7 pp 1285ndash1288 2016
[45] QHuang K Liu and ZWang ldquoLow-density arrays of circulantmatrices Rank and row-redundancy and QC-LDPC codesrdquoin Proceedings of the 2012 IEEE International Symposium onInformation Theory ISIT 2012 pp 3073ndash3077 USA July 2012
[46] H Xu D Feng C Sun and B Bai ldquoAlgebraic-based nonbinaryldpc codes with flexible field orders and code ratesrdquo ChinaCommunications vol 14 no 4 pp 111ndash119 2017
![IEEE ACCESS 1 A Flexible FPGA-Based Quasi-Cyclic LDPC Decoder · 2019-12-16 · LDPC codes to be iteratively decoded using a distributed low-complexity message-passing algorithm [2].](https://static.documents.pub/doc/80x56/5e7557354f01375926648a0f/ieee-access-1-a-flexible-fpga-based-quasi-cyclic-ldpc-decoder-2019-12-16-ldpc.jpg)
![A Massively Parallel Implementation of QC-LDPC Decoder ...gw2/pdf/sasp2011_gpu_ldpc_long.pdfQuasi-Cyclic LDPC (QC-LDPC) codes [1] have been widely used in many practical systems, such](https://static.documents.pub/doc/80x56/608577e15da5786347664f4c/a-massively-parallel-implementation-of-qc-ldpc-decoder-gw2pdfsasp2011gpuldpclongpdf.jpg)
