Bennatan2004.pdf

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 3, MARCH 2004 417

On the Application of LDPC Codes to ArbitraryDiscrete-Memoryless Channels

Amir Bennatan, Student Member, IEEE, and David Burshtein, Senior Member, IEEE

Abstract—We discuss three structures of modified low-densityparity-check (LDPC) code ensembles designed for transmissionover arbitrary discrete memoryless channels. The first structure isbased on the well-known binary LDPC codes following construc-tions proposed by Gallager and McEliece, the second is based onLDPC codes of arbitrary ( -ary) alphabets employing modulo-addition, as presented by Gallager, and the third is based on LDPCcodes defined over the field GF ( ). All structures are obtained byapplying a quantization mapping on a coset LDPC ensemble. Wepresent tools for the analysis of nonbinary codes and show that allconfigurations, under maximum-likelihood (ML) decoding, arecapable of reliable communication at rates arbitrarily close to thecapacity of any discrete memoryless channel. We discuss practicaliterative decoding of our structures and present simulation resultsfor the additive white Gaussian noise (AWGN) channel confirmingthe effectiveness of the codes.

Index Terms— -ary low-density parity check (LDPC), beliefpropagation, coset codes, iterative decoding, LDPC codes, turbocodes.

I. INTRODUCTION

LOW-density parity-check (LDPC) codes and iterative de-coding algorithms were proposed by Gallager [12] four

decades ago, but were relatively ignored until the introductionof turbo codes by Berrou et al. [2] in 1993. In fact, there is aclose resemblance between LDPC and turbo codes. Both arecharacterized by a LDPC matrix, and both can be decoded it-eratively using similar algorithms. Turbo-like codes have gen-erated a revolution in coding theory, by providing codes thathave capacity approaching transmission rates with practical it-erative decoding algorithms. Yet turbo-like codes are usuallyrestricted to single-user discrete-memoryless binary-input sym-metric-output channels.

Several methods for the adaptation of turbo-like codes tomore general channel models have been suggested. Wachs-mann et al. [31] have presented a multilevel coding schemeusing turbo codes as components. Robertson and Wörts [25]and Divsalar and Pollara [9] have presented turbo decodingmethods that replace binary constituent codes with trellis-codedmodulation (TCM) codes. Kavcic, Ma, Mitzenmacher, and

Manuscript received November 11, 2002; revised November 14, 2003. Thiswork was supported by the Israel Science Foundation under Grant 22/01-1 andby a fellowship from The Yitzhak and Chaya Weinstein Research Institute forSignal Processing at Tel-Aviv University. The material in this paper was pre-sented in part at the IEEE International Symposium on Information Theory,Yokohama, Japan, June/July 2003.

The authors are with the Department of Electrical Engineering–Sys-tems, Tel-Aviv University, Tel-Aviv, Ramat–Aviv 69978, Israel (e-mail:[email protected]; [email protected]).

Communicated by R. Koetter, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2004.824917

Varnica [16], [19], and [29] have suggested a method thatinvolves a concatenation of a set of LDPC codes as outer codeswith a matched-information rate (MIR) inner trellis code,which performs the function of shaping, an essential ingredientin approaching capacity [11].

In his book, Gallager [13], described a construction that canbe used to achieve capacity under maximum-likelihood (ML)decoding, on an arbitrary discrete-memoryless channel, usinga uniformly distributed parity-check matrix (i.e., each elementis set to uniformly, and independently of the other ele-ments). In his construction, Gallager used the coset-ensemble,i.e., the ensemble of all codes obtained by adding a fixed vectorto each of the codewords of any given code in the originalensemble. To obtain codes over large alphabets, Gallageralso proposed mapping groups of bits from a binary codeinto channel symbols. This mapping, which we have named“quantization,” can be designed to overcome the shapinggap to capacity. Gallager suggested these ideas in a generalcontext. Coset LDPC codes were also used by Kavcic et al.[15] in the context of intersymbol-interference (ISI) channels.McEliece [20] discussed the application of turbo-like codes toa wider range of channels, and proposed to use Gallager-stylemappings. A uniform mapping from -tuples to a constellationof signals was used in [28] and [5] in the contexts ofmultiple-input, multiple-output (MIMO) fading channels, andinterference mitigation at the transmitter, respectively.

In [12], Gallager proposed a generalization of standard binaryLDPC codes to arbitrary alphabet codes, along with a practicaliterative soft-decoding algorithm. Gallager presented an anal-ysis technique on the performance of the code that applies toa symmetric channel. LDPC codes over an arbitrary alphabetwere used by Davey and Mackay [8] in order to improve theperformance of standard binary LDPC codes. Hence, both con-structions target symmetric channels.

In this paper, we discuss three structures of modified LDPCcode ensembles designed for transmission over arbitrary (notnecessarily binary-input or symmetric-input) discrete memory-less channels. The first structure, called binary quantized coset(BQC) LDPC, is based on [13] and [20]. The second structureis a modification of [12], that utilizes cosets and a quantizationmap from the code symbol space to the channel symbol space.This structure, called modulo quantized coset (MQC) LDPC,assumes that the parity-check matrix elements remain binary,although the codewords are defined over a larger alphabet. Thethird structure is based on LDPC codes defined over Galoisfields GF and allows parity-check matrices over the entirerange of field elements. This last structure is called GF quan-tized coset (GQC) LDPC.

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418 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 3, MARCH 2004

Fig. 1. Encoding of BQC-LDPC codes.

We present tools for the analysis of nonbinary codes and de-rive upper bounds on the ML decoding error probability of thethree code structures. We then show that these configurations aregood in the sense that for regular ensembles with sufficientlylarge connectivity, a typical code in the ensemble enables re-liable communication at rates arbitrarily close to channel ca-pacity, on an arbitrary discrete-memoryless channel. Finally, wediscuss the practical iterative decoding algorithms of the variouscode structures and demonstrate their effectiveness.

Independently of our work, Erez and Miller [10] have re-cently examined the performance, under ML decoding, of stan-dard -ary LDPC codes over modulo-additive channels, in thecontext of lattice codes for additive channels. In this case, dueto the inherent symmetry of modulo-additive channels, neithercosets nor quantization mapping are required.

Our work is organized as follows. In Section II, we discussBQC-LDPC codes; in Section III, we discuss MQC-LDPCcodes; and in Section IV, we discuss GQC-LDPC codes.Section V discusses the iterative decoding of these structures. Italso presents simulation results and a comparison with existingmethods for transmission over the additive white Gaussiannoise (AWGN) channel. Section VI concludes the paper.

II. BINARY QUANTIZED COSET (BQC) CODES

We begin with a formal definition of coset-ensembles. Ourdefinition is slightly different from the one used by Gallager[13] and is more suitable for our analysis (see [26]).

Definition 1: Let be an ensemble of equal block-lengthbinary codes. The ensemble coset- (denoted ) is defined asthe output of the following two steps.

1. Generate the ensemble from by including, for eachcode in , codes generated by all possible permutations

on the order of bits in the codewords.

2. Generate ensemble from by including, for each codein , codes of the form ( de-

notes bitwise modulo- addition) for all possible vectors.

Given a code , we refer to as the coset vector andas the underlying code.

Note: The above definition is given in a general context.However, step 1 can be dropped when generating the coset-en-semble of LDPC codes, because LDPC code ensembles alreadycontain all codes obtained from a permutation of the order ofbits.

We proceed to formally define the concept of quantization.

Definition 2: Let be a rational probability assignment ofthe form , defined over the setA quantization associated with , of length

, is a mapping from the set of vectors tosuch that the number of vectors mapped to each

is .

Note that in Definition 2, the mapping of vectors to channelsymbols is, in general, not one-to-one. Hence the name “quanti-zation.” Such mapping enables the achievement of nonuniformdistributions , as required to approach the capacity of as-symetric channels. Applying the quantization to a vector oflength is performed by breaking the vector into subvec-tors of bits each, and applying to each of the subvectors.That is,

(1)

The quantization of a code is the code obtained from applyingthe quantization to each of the codewords. Likewise, the quanti-zation of an ensemble is an ensemble obtained by applying thequantization to each of the ensemble’s codes. A BQC ensembleis obtained from a binary code-ensemble by applying a quanti-zation to the corresponding coset-ensemble.

It is useful to model BQC-LDPC encoding as a sequence ofoperations, as shown in Fig. 1. An incoming message is encodedinto a codeword of the underlying LDPC code . The vector isthen added, and the quantization mapping applied. Finally, theresulting codeword is transmitted over the channel.

We now introduce the following notation, which is a general-ization of similar notation taken from literature covering binarycodes. For convenience, we formulate our results for the dis-crete-output case. The conversion to continuous output is im-mediate.

(2)

where denotes the transition probabilities of a givenchannel. Using the Cauchy–Schwartz inequality, it is easyto verify that and for all . This lastinequality becomes strict for nondegenerate quantizations andchannels, as explained in Appendix I-A. We assume theserequirements of nondegeneracy throughout the paper.

We now define

(3)

The analysis of the ML decoding properties of BQC codes isbased on the following theorem, which relates the decodingerror to the underlying code’s spectrum.

Theorem 1: Let be the transition probability assign-ment of a discrete memoryless channel with an input alphabet

. Let be a rate below channel capacity (the

BENNATAN AND BURSHTEIN: APPLICATION OF LDPC CODES TO DISCRETE-MEMORYLESS CHANNELS 419

rate being measured in bits per channel use), an arbitrarydistribution, and a quantization associated with oflength . Let be an ensemble of linear binary codes of length

and rate at least , and the ensemble average numberof words of weight in . Let be the BQC ensemble corre-sponding to .

Let be an arbitrary set of integers. Theensemble average error under ML decoding of , , satisfies

(4)

where is given by (3), is the random coding exponentas defined by Gallager in [13] for the input distribution

(5)

and is given by

(6)

, and .

The proof of this theorem is similar to the proof of Theorem 3and to proofs provided in [26] and [21]. A sketch of the proofis given in Appendix I-B. The proof of Theorem 3 is given indetail in Appendix II-A.

Theorem 1 is given in a context of general BQC codes. Wenow proceed to examine BQC-LDPC codes, constructed basedon the ensemble of -regular binary LDPC codes. It is con-venient to define a -regular binary LDPC code of length

by means of a bipartite -regular Tanner graph [27]. Thegraph has variable (left) nodes, corresponding to codewordbits, and check (right) nodes. A word is a codeword if ateach check node, the modulo- sum of all the bits at its adjacentvariable nodes is zero.

The following method, due to Luby et al. [18], is used to de-fine the ensemble of -regular binary LDPC codes. Eachvariable node is assigned sockets, and each check node is as-signed sockets. The variable sockets are matched (that is,connected using graph edges) to ( ) check sockets bymeans of randomly selected permutation of . Theensemble of LDPC codes consists of all the codes con-structed from all possible graph constellations.

The mapping from a bipartite graph to the parity-check matrixis performed by setting each matrix element , correspondingto the th check node and the th variable node, to the numberof edges connecting the two nodes modulo- (this definition isdesigned to account for the rare occurrence of parallel edgesbetween two nodes). The design rate of a regular LDPCcode is defined as . This value is alower bound on the true rate of the code, measured in bits perchannel use.

In [21], it is shown there exists such that the min-imum distance of a randomly selected code of a -reg-ular, -length, binary LDPC ensemble satisfies

(7)

The meaning of the above observation is that the performance ofthe ensemble is cluttered by a small number of codes (at worst).Removing these codes from the ensemble, we obtain an expur-gated ensemble with improved properties.

Formally, given an ensemble of -regular LDPC codesof length and an arbitrary number , we define theexpurgated ensemble as the ensemble obtained by removingall codes of minimum distance less than or equal to from

. If and is large enough, this does not reduce the sizeof the ensemble by a factor greater than . Thus, the expurgatedensemble average spectrum satisfies

(8)

We refer to the BQC ensemble corresponding to an expur-gated -regular ensemble as an expurgated BQC-LDPCensemble. We now use the above construction in the followingtheorem.


. Let be some rational positive number, and let, , , and be defined as in Theorem 1. Let

be two arbitrary numbers. Then, for and large enough,there exists a -regular expurgated BQC-LDPC ensemble

(containing all but a diminishingly small proportion of the-regular BQC-LDPC codes, as discussed above) of length

and design rate bits per channel use (the rate of each codein the ensemble is at least ), satisfying , suchthat the ML decoding error probability over satisfies

(9)

The proof of this theorem relies on results from [21] and isprovided in Appendix I-C.

Gallager [13] defined the exponent to be the max-imum, for each , of , evaluated over all possibleinput distributions . Let be the distribution thatattains this maximum for a given . By the nature of thequantization concept, we are restricted to input distributions

that assign rational probabilities of the form toeach . Nevertheless, by selectingto approximate , we obtain which approaches

equivalently close. for all and,thus, by appropriately selecting (recalling that ),we obtain BQC-LDPC codes that are capable of reliabletransmission (under ML decoding) at any rate below capacity.Furthermore, we have that the ensemble decoding error de-creases exponentially with . At rates approaching capacity,

approaches zero and hence (9) is dominated by therandom-coding error exponent.


Fig. 2. Upper bounds on the minimum required SNR for successful ML decoding of some BQC-LDPC codes over an AWGN channel. The solid line is theShannon limit.

Fig. 2 compares our bound on the threshold of several BQC-LDPC code ensembles, as a function of the signal-to-noise ratio(SNR), over the AWGN channel. The quantizations used are

uniformly spaced values ascending from

to (10)

The values of are listed in ascending binary order, e.g.,

(for the case of ). The quantizations were chosen suchthat a mean power constraint of is satisfied. Note that quanti-zation corresponds to equal-spaced 32-PAM transmission, ef-fectively representing transmission without quantization. Thus,the gap between codes and illustratesthe importance of quantization.

To obtain the threshold we use Theorem 1, employingmethods similar to the ones discussed in of [21, Sec. V]. Fora given -regular code and distribution we seek theminimal SNR that satisfies the following (for simplicity weassume that is even, and, therefore, ). We firstdetermine the maximal value of such that ,we then define ,and require that be positive (the maximumpoint yielding positive is evaluated using the mutualinformation as described in [13, Sec. 5.6]).

III. MODULO- QUANTIZED COSET (MQC) LDPC CODES

MQC-LDPC codes are based on modulo- LDPC codes. Thedefinition of -regular, modulo- LDPC codes is adaptedfrom its binary equivalent (provided in Section II) and is aslightly modified version of Gallager’s definition [12, Ch. 5].Bipartite -regular graphs are defined and constructed inthe same way as in Section II, although variable nodes areassociated with -ary symbols rather than bits. A -ary word

is a codeword if at each check node, the modulo- sum ofall symbols at its adjacent variable nodes is zero. The mappingfrom a bipartite graph to the parity-check matrix is performedby setting each matrix element , corresponding to the thcheck node and the th variable node, to the number of edgesconnecting the two nodes modulo- (we thus occasionallyobtain matrix elements of a nonbinary alphabet).

As in the binary case, the design rate of a regularmodulo- LDPC code is defined as .


Assuming that is prime, this value is a lower bound on thetrue rate of the code, measured in -ary symbols per channeluse. The prime assumption does not pose a problem, as thetheorems presented in this section generally require to beprime.

We proceed by giving the formal definitions of coset-ensem-bles and quantization for arbitrary-alphabet codes. Note that thedefinitions used here are slightly different to the ones used withBQC codes.

Definition 3: Let be an ensemble of equal block lengthcodes over the alphabet . The ensemble coset-(denoted ) is generated from by adding, for each code in

, codes of the form for all possible vectors.

Definition 4: Let be a rational probability assignment ofthe form , defined over the set

. A quantization associated with is amapping from a set to wherethe number of elements mapped to each is

.

A quantization is applied to a vector by applying the abovemapping to each of its elements. As in the binary case, the quan-tization of a code is the code obtained from applying the quan-tization to each of the codewords. The quantization of an en-semble is an ensemble obtained by applying the quantization toeach of the ensemble’s codes. As with BQC codes, a MQC en-semble is obtained from a modulo- code ensemble by applyinga quantization to the corresponding coset-ensemble.

Analysis of ML decoding of binary-based BQC codes is fo-cused around the weight distribution of codewords. With -ary-based MQC codes, weight is replaced by the concept of type(note that in [7, Sec. 12.1] the type is defined differently, as anormalized value).

Definition 5: The type of a vectoris a -dimensional vector of integers such that

is the number of occurrences of the symbol in . We denotethe set of all possible types by .

The spectrum of a -ary code is defined in a manner similarto that of binary codes.

Definition 6: The spectrum of a code is defined as, where is the number of words of type in .

We now introduce the notation (not tobe confused with the similar definition of in (2)) defined by

(11)

where denotes the transition probabilities of a givenchannel, and is a quantization. Using the Cauchy–Schwartzinequality, it is easy to verify that and for all

. As in the case of BQC codes, the last inequality becomesstrict for nondegenerate quantizations and channels (defined asin Appendix I-A, replacing with ).

Given a type , we define

The -ary (uniform distribution) random coding ensemble iscreated by randomly selecting each codeword and each code-word symbol independently and with uniform probability. Theensemble average spectrum (average number of codewords oftype ) of the random coding ensemble is given by

where is the number of codewords in each code and is thecodeword length.

The importance of the random-coding spectrum is given bythe following theorem.


. Let be a rate below channel capacity (therate being measured in -ary symbols per channel use), anarbitrary distribution, and a quantization associated with

over the alphabet . Let be an ensemble oflinear modulo- codes of length and rate at least , andthe ensemble average of the number of words of type in . Let

be the MQC ensemble corresponding to .Let be a set of types. Then the ensemble average error

under ML decoding of , , satisfies

(12)

where is defined as in (11), is given by (5), and isgiven by

(13)

, where is the type of the all-zeros word, and.

The proof of this theorem is provided in Appendix II-A.The normalized ensemble spectrum of -ary codes is de-

fined in a manner similar to that of binary codes

(14)

where denotes a -dimensional vector of rational numbers sat-isfying . Throughout the rest of this paper, weadopt the convention that the base of the function is always .

The normalized spectrum of the random coding ensemble isgiven by

where denotes the entropy function, and is the code rate(in -ary symbols per channel use). The normalized spectrum ofmodulo- LDPC codes is given by the following theorem.


Fig. 3. J for � = 0:2.

Theorem 4: The asymptotic normalized ensemble spectrumof a -regular modulo- LDPC code over the alphabet

is given by

(15)

where , is given by

, and is given by

(16)

The proof of this theorem is provided in Appendix II-C.We now show that the modulo- LDPC normalized spectrum

is upper-bounded arbitrarily closely by the random-coding nor-malized spectrum.

Theorem 5: Let be a prime number, let be anarbitrarily chosen number, and

(17)Let be a given rational positive number and be therandom-coding normalized spectrum corresponding to the rate

. Then for any there exists a number such that

(18)

for all , and all , satisfying and .

The proof of this theorem is provided in Appendix II-D.Fig. 3 presents the set for a ternary alphabet. The andaxes represent variables with being implied by the

relation . A triangle outlines the region of validvalues for (that is, and ).

Fig. 4 presents the normalized spectrums of a ternary -regular LDPC code (the upper surface) and of the ternaryrate– random-coding ensemble (the lower surface). Thenormalized spectra are plotted as functions of parameters

, with .

Theorem 5 provides uniform convergence only in a subset ofthe space of valid values of . This means that the maximum(see (13)), evaluated over all values of , may be large regardlessof the values of and . The solution to this problem follows inlines analogous to the binary case. We begin with the followinglemma.

Lemma 1: Let be a word of weight ( nonzero ele-ments). We now bound the probability of being a codewordof a randomly selected code of the -regular modulo-ensemble .

1) If then

(19)

where is the number of check nodes in theparity-check matrix of a -regular code and de-notes the largest integer smaller than or equal to .

2) For all , assuming prime

(20)

where , is given by

(21)

and is some constant dependent on alone satisfying.

The proof of this lemma is provided in Appendix II-E.We now build on this lemma to examine the probability of

there being any low-weight words in a randomly selected codefor an LDPC ensemble.

Theorem 6: Let be fixed, , and assumeis prime. Then there exists , dependent on and alone,such that

(22)

where is the minimum distance of a randomly selected-regular modulo- code of length .1

The proof of this theorem is provided in Appendix II-F.Given an ensemble of -regular LDPC codes of lengthand an arbitrary number , we define the expurgated

ensemble as the ensemble obtained by removing all codes ofminimum distance less than or equal to from . If(where is given by Theorem 6) and is large enough, thisdoes not reduce the size of the ensemble by a factor greater than

. Thus, the expurgated ensemble average spectrum satisfies

(23)

where is the number of nonzero elements in a word oftype . That is, . We refer to theMQC ensemble corresponding to an expurgated -regularensemble as an expurgated MQC-LDPC ensemble.

1In fact, it can be shown that fixing R = 1 � c=d and letting c; d ! 1,the minimum distance of a randomly selected code is, with high probability,lower-bounded by a value arbitrarily close to the Varshamov–Gilbert bound.


Fig. 4. Comparison of the normalized spectrums of the ternary (3; 6)-regular LDPC code ensemble and the ternary rate-1=2 random-coding ensemble.


. Assume is prime. Let be some rationalpositive number, and let , , and be defined as inTheorem 3. Let be two arbitrary numbers. Thenfor and large enough there exists a -regular expur-gated MQC-LDPC ensemble (containing all but a diminish-ingly small proportion of the -regular MQC-LDPC codes,as shown in Theorem 6) of length and design rate , satis-fying , such that the ML decoding error probability

over satisfies

(24)

The proof of this theorem is provided in Appendix II-G.Applying the same arguments as with BQC-LDPC codes,

we obtain that MQC-LDPC codes can be designed for reliabletransmission (under ML decoding) at any rate below capacity.Furthermore, at rates approaching capacity, (24) is dominatedby the random-coding error exponent.

Producing bounds for individual MQC-LDPC code ensem-bles, similar to those provided in Figs. 2 and 5, is difficult dueto the numerical complexity of evaluating (13), for large , oversets given by (17).

Note: In our constructions, we have restricted ourselves toprime values of . In Appendix II-H we show that this restrictionis necessary, at least for values of that are multiples of .

IV. QUANTIZED COSET LDPC CODES OVER GF (GQC)

GQC-LDPC codes are based on LDPC codes defined overGF . We therefore begin by extending the definition of LDPC

codes to a finite field GF .2 In our definition of modulo-LDPC codes in Section III, the enlargement of the code al-phabet size did not extend to the parity-check matrix. The parity-check matrix was designed to contain binary digits alone (withrare exceptions involving parallel edges). We shall see in Ap-pendix II-H that for nonprime , this construction results incodes that are bounded away from the random-coding spec-trum. The ideas presented in Appendix II-H are easily extendedto ensembles over GF for . We therefore definethe GF parity-check matrix differently, employing elementsfrom the entire GF field.

Bipartite -regular graphs for LDPC codes over GFare constructed in the same way as described in Sections II andIII, with the following addition: At each edge , a random,uniformly distributed label GF is selected. Aword is a codeword if at each check node the followingequation holds:

where is the set of variable nodes adjacent to .The mapping from the bipartite graph to the parity-check

matrix proceeds as follows. Element in the matrix, cor-responding to the th check node and the th variable node, isset to the GF sum of all labels corresponding to edgesconnecting the two nodes. As before, the rate of each code islower-bounded by the design rate -arysymbols per channel use.

2Galois fields GF (q) exist for values of q satisfying q = p where p is primeand m is an arbitrary positive integer. See [3].


Fig. 5. Upper bounds on the minimum required SNR for successful ML decoding of some GQC-LDPC codes over an AWGN channel. The solid line is theShannon limit.

GF coset-ensembles and quantization mappings are de-fined in the same way as with modulo- codes. Thus, we obtainGF quantized coset (GQC) code-ensembles.

As with MQC codes, analysis of ML decoding properties ofGQC LDPC codes involves the types rather than the weightsof codewords. Nevertheless, the analysis is simplified using thefollowing lemma.

Lemma 2: Let and be two equal-weight words.The probabilities of the words belonging to a randomly selectedcode from a GF , -regular ensemble satisfy

The proof of this lemma relies on the following two observa-tions. First, by the symmetry of the construction of the LDPCensemble, any reordering of a word’s symbols has no effect onthe probability of the word belonging to a randomly selectedcode. Second, if we replace a nonzero symbol by another ,we can match any code with a unique code suchthat the modified word belongs to if and only if the originalword belongs to . The code is constructed by modifyingthe labels on the edges adjacent to the corresponding variablenode using (evaluated over GF ).

We have now obtained that the probability of a word be-longing to a randomly selected code is dependent on its weightalone, and not on its type. We use this fact in the following the-orem, to produce a convenient expression for the normalizedspectrum of LDPC codes over GF .

Theorem 8: The asymptotic normalized ensemble spectrumof a -regular LDPC code over GF is given by

(25)

where and is given by

(26)

Note that in (25) the parameter is implied by the relationand, therefore, the right-hand side of the equation

is, in fact, a function of alone. The proof of the theorem isprovided in Appendix III-A.

Theorems 3, 5, 6, and Lemma 1 carry over from MQC-LDPCto GQC-LDPC codes, with minor modifications. In Theorem3, modulo- addition is replaced by addition over GF . InTheorem 5, the requirement is added and the definitionof is replaced by

(27)

The proof is similar to the case of MQC-LDPC codes, settingin (25) to upper-bound . In

Lemma 1, (20) is replaced by


where and is given by

Finally, Theorem 6 carries over unchanged from MQC-LDPCcodes.

The definition of expurgated GQC-LDPC ensembles is iden-tical to the equivalent MQC-LDPC definition. We now state themain theorem of this section (which is similar to Theorem 7).


. Suppose . Let be some rational positivenumber, and let , , and be defined as in Theorem3. Let be an arbitrary number. Then for and largeenough there exists a -regular expurgated GQC-LDPC en-semble (containing all but a diminishingly small proportionof the -regular GQC-LDPC codes, as shown in Theorem6) of length and design rate , satisfying , suchthat the ML decoding error probability over satisfies

(28)

The proof of this theorem follows in the lines of Theorem 7,replacing (78) with

(29)

The difference between (28) and (24) results from the largerspan of the set defined in (27) in comparison with the onedefined in Theorem 5. Thus, we are able to define such thatthe first term in (12) disappears.

Applying the same arguments as with BQC-LDPC codes andMQC-LDPC codes, we obtain that GQC-LDPC codes can bedesigned for reliable transmission (under ML decoding) at anyrate below capacity. Furthermore, the above theorem guaranteesthat at any rate, the decoding error exponent for GQC-LDPCcodes asymptotically approaches the random-coding exponent,thus outperforming our bounds for both the BQC-LDPC andMQC-LDPC ensembles.

Fig. 5 compares several GQC-LDPC code ensembles over theAWGN channel. The quantizations used are the same as thoseused for Fig. 2 and are given by (10) (using the representationof elements of GF by -dimensional binary vectors [3]).To obtain these bounds, we again use methods similar to theones discussed above for BQC-LDPC codes (here we seek themaximum such that for all having

and then define as in (29)).

V. ITERATIVE DECODING

Our analysis so far has focused on the desirable properties ofthe proposed codes under optimal ML decoding. In this section,we demonstrate that the various codes show favorable perfor-mance also under practical iterative decoding.

The BQC-LDPC belief propagation decoder is based on thewell-known LDPC decoder, with differences involving the ad-dition of symbol nodes alongside variable and check nodes, de-rived from a factor graph representation by McEliece [20].

Fig. 6. Diagram of the bipartite graph of a (2; 3)-regular BQC-LDPC codewith T = 3.

The decoding process attempts to recover , the codeword ofthe underlying LDPC code. Fig. 6 presents an example of the bi-partite graph for a -regular BQC-LDPC code with .Variable nodes and check nodes are defined in a manner similarto that of standard LDPC codes. Symbol nodes correspond tothe symbols of the quantized codeword (1). Each symbol nodeis connected to the variable nodes that make up the subvectorthat is mapped to the symbol. Decoding consists of alternatingrightbound and leftbound iterations. In a rightbound iteration,messages are sent from variable nodes to symbol nodes andcheck nodes. In a leftbound iteration, the opposite occurs. Un-like the standard LDPC belief-propagation decoder, the channeloutput “resides” at the symbol nodes, rather than at the variablenodes. The use of the coset vector is easily accounted for at thesymbol nodes.

The MQC-LDPC and GQC-LDPC belief propagation de-coders are modified versions of the belief propagation decoderintroduced by Gallager in [12] for arbitrary alphabet LDPCcodes, employing -dimensional vector messages. The mod-ifications, easily implemented at the variable nodes, accountfor the addition of the coset vector at the transmitter andfor the use of quantization. Efficient implementation of beliefpropagation decoding of arbitrary-alphabet LDPC codes isdiscussed by Davey and MacKay [8] and by Richardson andUrbanke [23]. The ideas in [8] are suggested in the context ofLDPC codes over GF but also apply to codes employingmodulo- arithmetic. The method discussed in [23] suggestsusing the discrete Fourier transform (DFT) (the -dimensionalDFT for codes over GF ) to reduce complexity. This isparticularly useful for codes defined over GF , since thenthe multiplications are eliminated.

An important property of quantized coset LDPC codes (BQC,MQC, and GQC) is that for any given bipartite graph, the prob-ability of decoding error, averaged over all possible values ofthe coset vector , is independent of the transmitted codewordof the underlying LDPC code (as observed by Kavcic et al. [15]in the context of coset LDPC codes for ISI channels). This fa-cilitates the extension of the density evolution analysis method(see [23]) to quantized coset codes.

BQC-LDPC density evolution evaluates the density of mes-sages originating from symbol nodes using Monte Carlo simu-lations. Density evolution for MQC- and GQC-LDPC codes is


complicated by to the exponential amount of memory requiredto store the probability densities of -dimensional messages. Apossible solution to this problem is to rely on Monte Carlo sim-ulations to evolve the densities.

A. Simulation Results for BQC-LDPC Codes

In this paper, we have generally confined our discussion toLDPC codes based on regular graphs. Nevertheless, as withstandard LDPC codes, the best BQC-LDPC codes under be-lief-propagation decoding are based on irregular graphs as in-troduced by Luby et al. [18]. In this section, we therefore focuson irregular codes.

A major element in generating good BQC-LDPC codes is thedesign of good quantizations. In our analysis of ML decoding,we focused on the probability assignment associated withthe quantizations. A good probability assignment for memory-less AWGN channels is generally designed to approximate aGaussian distribution (see [30] and [31]). Iterative decoding,however, is sensitive to the particular mapping .

A key observation in the design of BQC-LDPC quantizationsis that the degree of error protection provided to different bitsthat are mapped to the same channel symbol is not identical.Useful figures of merit are the following values (adapted from[24, Sec. III.E], replacing the log-likelihood messages of [24]with plain likelihood messages):

(30)

where is a random variable corresponding to the leftboundmessage from the th position of a symbol node, assuming thetransmitted symbol was zero, in the first iteration of belief prop-agation. It is easy to verify that the following relation holds:

Quantizations rendering a low for a particular produce astronger leftbound message at the corresponding position, atthe first iteration. Given a particular distribution , differentquantizations associated with produce different sets

. In keeping with the experience available for standardLDPC codes, simulation results indicate that good quantizationsfavor an irregular set of , meaning that some values ofthe set are allowed to be larger than others. In the design of edgedistributions, to further increase irregularity, it is useful to con-sider not only the fraction of edges of given left degree butrather , accounting for the left variable node’s symbol nodeposition.

In this paper, we have employed only rudimentary methodsof designing BQC-LDPC edge distributions. Table I presents theedge distributions of a rate– (measured in bits per channel use)BQC-LDPC code for the AWGN channel. The following length

quantization was used with 8-PAM signaling .

TABLE ILEFT EDGE DISTRIBUTION FOR A RATE-2 BQC-LDPC CODE WITHIN 1.1 dB

OF THE SHANNON LIMIT. THE RIGHT EDGE DISTRIBUTION IS GIVEN BY

� = 0:25, � = 0:75

TABLE IILEFT EDGE DISTRIBUTION FOR A RATE-2 MQC-LDPC CODE WITHIN 0.9 dB

OF THE SHANNON LIMIT. THE RIGHT EDGE DISTRIBUTION IS GIVEN

BY � = 0:5, � = 0:5

Applying the notation of (10), we obtain that a simple assign-ment of values in an ascending order renders a good quantization

The SNR threshold (determined by simulations) is 1.1 dB awayfrom the Shannon limit (which is 11.76 dB at rate ) at a blocklength of 200 000 bits or 50 000 symbols and a bit-error rate of about . Decoding typically requires 100–150iterations to converge. Note that the ML decoding threshold ofa random code, constructed using the distribution asso-ciated with , is 11.99 dB (evaluated using the mutual in-formation as described in [13, Sec. 5.6]). Thus, the gap to therandom-coding limit is 0.87 dB. The edge distributions were de-signed based on a rate- binary LDPC code suggested by [24],using trial-and-error to design while keeping the marginaldistribution fixed.

B. Simulation Results for MQC-LDPC Codes

The MQC equivalent of (30) are the values givenby (11). As with BQC-LDPC codes, MQC-LDPC codes favorquantizations rendering an irregular set (as indicatedby simulation results).

In [24] and [6], methods for the design of edge distributionsare presented that use singleton error probabilities produced bydensity evolution and iterative application of linear program-ming. In this work, we have used a similar method, replacingthe probabilities produced by density evolution with results ofMonte Carlo simulations. An additional improvement was ob-tained by replacing singleton error probabilities by the func-tional ( denoting -ary entropy).

Table II presents the edge distributions of a rate– (mea-sured again in bits per channel use) MQC-LDPC code for theAWGN channel. The code alphabet size is , and -PAMsignaling was used , with the quantization listedbelow. The values of are listed in ascending order, i.e.,

. The quantization values were selected


TABLE IIILEFT EDGE DISTRIBUTION FOR A RATE-2 GQC-LDPC CODE WITHIN 0.65 dB

OF THE SHANNON LIMIT. THE RIGHT EDGE DISTRIBUTION IS GIVEN

BY � = 0:5, � = 0:5

to approximate a Gaussian distribution, and their order wasdetermined by trial-and-error

At an SNR 0.9 dB away from the Shannon limit, the symbolerror rate was (50 Monte Carlo simulations). The gapto the ML decoding threshold of a random code, constructedusing the distribution associated with , is 0.7 dB.The block length used was 50 000 symbols or204 373 bits. Decoding typically required 100–150 iterations toconverge.

C. Simulation Results for GQC-LDPC Codes

In contrast to BQC-LDPC and MQC-LDPC codes, GQC-LDPC appear resilient to the ordering of values in the quantiza-tions used. This is the result of the use of random labels, whichinfer a permutation on the rightbound and leftbound messagesat check nodes.

Table III presents the edge distributions of a rate– (mea-sured again in bits per channel use) GQC-LDPC code for theAWGN channel. The methods used to design the codes are thesame as those described above for MQC-LDPC codes. Thecode alphabet size is , and -PAM signaling was used

, with the quantization listed below. The values ofare listed in ascending order (using the representation

of elements of GF by -dimensional binary vectors[3]). The quantization values were selected to approximate aGaussian distribution

At an SNR 0.65 dB away from the Shannon limit, the symbolerror rate was less than (100 Monte Carlo simulations).The gap to the ML decoding threshold of a random code, con-structed using the distribution associated with , is0.4 dB. The block length used was 50 000 symbols or

200 000 bits. Decoding typically required 100–150 itera-tions to converge.

D. Comparison With Existing Methods

Wachsmann et al. [31], present reliable transmission usingmultilevel codes with turbo code constituent codes, at approx-imately 1 dB of the Shannon limit at rate 1 bit per dimension.Multilevel coding is similar to BQC quantization mapping, in

that multiple bits (each from a separate binary code) are com-bined to produce channel symbols. However, the division ofthe code into independent subcodes and the hard decisions per-formed during multistage decoding, are potentially suboptimal.

Robertson and Wörts [25], using turbo codes with TCM con-stituent codes, report several results that include reliable trans-mission within 0.8 dB of the Shannon limit at rate 1 bit per di-mension.

Kavcic, Ma, Mitzenmacher, and Varnica [16], [19], and [29]present reliable transmission at 0.74 dB of the Shannon limit atrate 2.5 bits per dimension. Their method has several similar-ities to BQC-LDPC codes. As in multilevel schemes, multiplebits from the outer codes are combined and mapped into channelsymbols in a manner similar to BQC quantizations. Quantiza-tion mapping can be viewed as a special case of a one-stateMIR trellis code, having parallel branches connecting everytwo subsequent states of the trellis. Furthermore, the irregularLDPC construction method of [19] and [29] is similar to themethod of Section V-A. In [19] and [29], the edge distribu-tions for the outer LDPC codes are optimized separately, andthe resulting codes are interleaved into one LDPC code. Simi-larly, BQC-LDPC construction distinguishes between variablenodes of different symbol node position. However, during theconstruction of the BQC-LDPC bipartite graph, no distinction ismade between variable nodes in the sense that all variable nodesare allowed to connect to the same check nodes (see Section II).This is in contrast to the interleaved LDPC codes of [19] and[29], where variable nodes originating from different subcodesare connected to distinct sets of check nodes.

VI. CONCLUSION

The codes presented in this paper provide a simple approachto the problem of applying LDPC codes to arbitrary discretememoryless channels. Quantization mapping (based on ideas byGallager [13] and McEliece [20]) has enabled the adaptation ofLDPC codes to channels where the capacity-achieving sourcedistribution is nonuniform. It is thus a valuable method of over-coming the shaping gap to capacity. The addition of a randomcoset vector (based on ideas by Gallager [13] and Kavcic et al.[15]) is a crucial ingredient for rigorous analysis.

Our focus in this paper has been on ML decoding. We haveshown that all three code configurations presented, under MLdecoding, are capable of approaching capacity arbitrarily close.Our analysis of MQC and GQC codes has relied on generaliza-tions of concepts useful with binary codes, like code spectrumand expurgated ensembles.

We have also demonstrated the performance of practical itera-tive decoding. The simple structure of the codes presented lendsitself to the design of conceptually simple decoders, which arebased entirely on the well-established concepts of iterative be-lief-propagation (e.g., [20], [22], [12]). We have presented sim-ulation results of promising performance within 0.65 dB of theShannon limit at a rate of 2 bits per channel use. We are cur-rently working on an analysis of iterative decoding of quantizedcoset LDPC codes [1].


APPENDIX IPROOFS FOR SECTION II

A. Proof of for for NondegenerateQuantizations and Channels

We define a quantization to be nondegenerate if there ex-ists no integer such that is an integer mul-tiple of for all (such a quantization could be replaced by asimpler quantization over an alphabet of size that wouldequally attain input distribution ). A channel is nondegen-erate if there exist no values such that

for all . By inspecting when theCauchy–Schwartz inequality becomes an equality, we see that

if and only iffor all and . Hence, by the channel nondegeneracy assump-tion, , for all . This contradicts the quantiza-tion nondegeneracy assumption, since the number of elementsin that are mapped to each channel symbol is some in-teger multiple of the additive order of (which is two).

B. Sketch of Proof for Theorem 1

The proofs of Theorems 1 and 3 are similar. In this paper, wehave selected to elaborate on Theorem 3, as the gap between itsproof and proofs provided in [26] and [21] is greater. Thus, weconcentrate here only on the differences between the two.

As in Appendix II-A, we define and (and equiv-

alently and ) as

was transmitted

where is a code of ensemble , defined as in Definition 1. Weobtain (using the union bound)

(31)

We proceed to bound both elements of the sum.

1) As in Appendix II-A we obtain

Recalling (1) and defining

we obtain the following identity:

Each element of the vector is now dependent only onthe transmitted symbol at position . We thereforeobtain

Using the above result, we now examine , theprobability of error for a fixed parent code (obtainedby Step 1 of Definition 1), averaged over all possiblevalues of (see the first equation at the bottom of thepage). and are defined in a manner similar to thedefinition of . Letting and

we obtain (32) at the bottom of the page, whereis defined as in (2). Defining we

observe that at least of the componentsare not equal to , and hence, recalling (3)

where is the spectrum of code . Clearly, this boundcan equally be applied to , abandoning the assump-tion of a fixed . The average spectrum of ensembleis equal to that of our original ensemble , and thereforewe obtain

(33)

2) The methods used to bound the second element of (31)are similar to those used in Appendix II-A. The boundobtained is

(34)

Combining (31), (33), and (34) we obtain our desired re-sult of (4).

(32)


C. Proof of Theorem 2

Let be defined as in (7). Let be some arbitrary numbersmaller than that will be determined later. Let be the un-derlying expurgated LDPC ensemble of , obtained by expur-gating codes with minimum distance less than or equal to .We use Theorem 1 to bound .

Defining

we have (using (4))

(35)

We now examine both elements of the above sum.

1) Given that all codes in the expurgated ensemble have min-imum distance greater than , we obtain thatfor all . We therefore turn to examine .

A desirable effect of expurgation is that it also reducesthe number of words having large . Assume and aretwo codewords of a code , such that their weightsatisfies . The weight of the codeword

satisfies , contradicting the con-struction of the expurgated ensemble. Hence, cannotcontain more than one codeword having . Letting

denote the spectrum of code , we have

Clearly, this bound carries over to the average spectrum.Selecting so that we obtain

(36)

2) Examining the proof by [21, Appendix E], we obtain

Therefore, there exist positive integers and suchthat for , , satisfying , , and

the following inequality holds:

(37)

Combining (35)–(37) we obtain our desired result of (9).

APPENDIX IIPROOFS FOR SECTION III

A. Proof of Theorem 3

The proof of this theorem is a generalization of the proofsavailable in [21] and [26], from binary-input symmetric outputto arbitrary channels.

All codes in have the form for somecode of . We can therefore write

where

was transmitted

We define and (and, equivalently, and )as

type was transmitted

(38)

and obtain (using the union bound)

(39)

We now proceed to bound both elements of the sum.

1) We first bound (defined above), the error proba-bility for a fixed index and code . This is the proba-bility that some codeword such that type

is more likely to have produced the observation thanthe true

type was transmitted

where is defined as shown in the equation at thebottom of the page. Therefore,

otherwise.


Each code has the form for someparent code and a vector . We now use the aboveresult to bound , the probability of error for a fixedparent code , averaged over all possible values of

The elements of the random vector are indepen-dent and identically distributed (i.i.d.) random variables

. Therefore,

Letting and we obtain

where is defined as in (11). Lettingtype , we have

Clearly, this bound can equally be applied to , aban-doning the assumption of a fixed . Finally, we bound

(40)

2) We now bound . For this we define a set of auxiliaryensembles.

a) Ensemble is generated from by adding, for eachcode in , the codes generated by all possiblepermutations on the order of codewords within thecode.

b) Ensemble is generated from by adding, foreach code in , the codes generated by all pos-

sible permutations on the order of symbols withinthe code.

c) Ensemble is generated from by adding,for each code in , codes of the form

for all possible vectors.

Examining the above construction, it is easy to verify thatany reordering of the steps has no impact on the finalresult. Therefore, ensemble can be obtained from byemploying Steps 2a) and 2b).

We also define ensemble as the quantization of .can equivalently be obtained employing the above Steps2a) and 2b) on ensemble . Therefore, , definedbased on (38), is identical to a similarly defined ,evaluated over ensemble .

We proceed to examine

(41)

where is defined as the probability, for a fixedand index , assuming , that a codeword of

the form , where shouldbe more likely than the transmitted . The randomspace here consists of a random selection of the codefrom . Therefore,

type

type

Letting be an arbitrary number, we have bythe union bound (extended as in [13])

Employing Lemma 3, which is provided in Ap-pendix II-B, we obtain


Letting , we have

where is some arbitrary number. Lettingwe have

(42)

Using (41) and (42) we obtain

Employing Lemma 3 once more, we obtain

The remainder of the proof follows in direct lines as in[13, Theorem 5.6.1]. Abandoning the assumption of afixed , and recalling that , we obtain

(43)

Combining (39), (40), and (43) we obtain our desired result of(12).

B. Statement and Proof of Lemma 3

The following lemma is used in the proof of Theorem 3 andrelies on the notation introduced there.

Lemma 3: Let and letbe two distinct indexes. Let and be the respective code-words of a randomly selected code of . Then

1) .2) If type then

(44)

where is defined by (13).Proof: Let be a randomly selected code from . We first

fix the ancestor code and investigate the relevant probabili-

ties. We define the following random variables: denotes the“parent” code from which was generated. Likewise, de-notes the parent of , and denotes the parent of .

Let be an arbitrary nonzero word. We now examine thecodewords of

otherwise

where is the number of codewords in . Recalling thatis of rate at least , we have that and, hence,

.We now examine

where is the type of . The last equation holds because, givena uniform random selection of the symbol permutation , theprobability of being a codeword of is equal to thefraction of -type codewords within the entire set of -typewords.

Examining we have

where is now the type of .We finally abandon the assumption of a fixed and obtain

If is in , we obtain from (13) the desired result of (44), andthus complete the proof of the lemma.

C. Proof of Theorem 4

To prove this theorem, we first quote the following notationand theorem of Burshtein and Miller [4] (see also [14]). Givena multinomial , the coefficient of is de-


noted3 . The theorem is stated for thetwo-dimensional case but is equally valid for higher dimen-sions (we present a truncated version of the theorem actuallydiscussed in [4]).

Theorem 10: Let be some rational number and letbe a function such that is a multinomial with

nonnegative coefficients. Let and be some rationalnumbers and let be the series of all indexes such thatis an integer and . Then

(45)

and

(46)

We now use this theorem to prove Theorem 4. The proof fol-lows the lines of a similar proof for binary codes in [4].

We first calculate the asymptotic spectrum for satisfyingfor all .

Given a type and

let be an indicator random variable equal to if the thword of type (by some arbitrary ordering of typewords) is acodeword of a drawn code, and otherwise. Then

Therefore,

(47)

the final equality resulting from the symmetry of the ensembleconstruction. Thus, for all

(48)

3The same notation bxc is used to denote the largest integer smaller than orequal to x. The distinction between the two meanings is to be made based onthe context of the discussion.

We now examine . Let be a word of type .Each assignment of edges by a random permutation infers asymbol “coloring” on each of the check node sockets, using

colors of type . The total numberof colorings is

(49)

Given an assignment of graph edges, is a codeword if at eachcheck node, the modulo- sum of all adjacent variable nodes iszero. The number of valid assignments (assignments rendering

a codeword), is determined using an enumerating function

(50)

where is given by the first equation at the bottom of thepage. We now have

(51)

Using (49) we have

(52)

Using (50) and Theorem 10 we have

(53)

Combining (48), (51), (52), and (53) we obtain

To adapt the proof to having for some , we modify theexpression for as follows. Assuming (without loss ofgenerality) that for all , and for all ,we obtain

where is given by the second equation at thebottom of the page. Following the line of previous development,we obtain


We now obtain (16). The development follows the lines of asimilar development in [12, Theorem 5.1]. Given , we definefor

and

(54)

Therefore, we obtain the expression for and subsequently(55) (both at the bottom of the page). Combining (54) with (55)we obtain (16).

D. Proof of Theorem 5

From (15), we obtain, selecting

(56)

We now bound for ( being defined in (17)).We introduce the following standard notation, borrowed fromliterature covering the DFT:

(57)

(58)

Incorporating this notation into (16), we now have

(59)

Let .

• Using we have that there exists at least onesatisfying .

• Using , we obtain that there exists at least onesatisfying .

We now bound for all

(60)

For

(61)

We separately treat both elements of the above sum

(62)

Using , , and and usingthe fact that is prime, we obtain that and have nocommon divisors. Therefore, . Defining as

(63)

we have

therefore,

Taking the square root of both sides of the above equation, weobtain

(64)

where is some positive constant smaller than , dependent onand but independent of and .

and

(55)


Combining (61), (62), and (64) we obtain

(65)

where , like , is some positive constant smaller than , de-pendent on and but independent of and .

Recalling (59)

using (60) and (65) we obtain

and, therefore, approaches with uniformly on . Wenow obtain

where the above bound is obtained uniformly over . Fi-nally, combining this result with (56) we obtain our desired re-sult of (18).

E. Proof of Lemma 1

1) The proof of (19) is similar to the proof of Lemma 2 in[21]. We concentrate here only on the differences betweenthe two, resulting from the enlargement of the alphabetsize. The mapping between a bipartite graph and a matrix

, defined in [21], is extended so that an elementof the matrix is set to if the corresponding left vertex(variable node) is nonzero, and to otherwise. Therefore,although our alphabet size is in general larger than ,we still restrict the ensemble of matrices to binaryvalues.

Unlike [21], applying the above mapping to a valid-ary codeword does not necessarily produce a matrix

whose columns are all of an even weight. Therefore, therequirement that be even does not hold. However, thematrix must still satisfy the condition that each of itspopulated columns have a weight of at least . Hence, thenumber of populated columns cannot exceed . Theremainder of the proof follows in direct lines as in [21].

2) We now turn to the proof of (20). Let be the type of. As we have seen in Appendix II-C, (49), and (50)

(66)

To bound the numerator, we use methods similar to thoseused to obtain (53), applying the bound (45) rather thanthe limit of (46). We obtain

(67)

To lower-bound the denominator, we use the well-knownbound (e.g., [7, Theorem 12.1.3])

(68)

Combining (66), (67), and (68) we obtain

(69)

Using arguments similar to those used to obtain (56), weobtain

(70)

Recalling that is the type of word and given thatis the number of nonzero elements in , we

obtain . We now bound for values ofsatisfying , , and

. To do this, we use methods similar to thoseused in Appendix II-D. Employing the notation of (57)and (58), we bound for

We first examine the elements of the second sum on theright-hand side of the above equation. As in the proofof Theorem 5, we have for all .Defining as in (63), we obtain

We now have


The first sum in the last equation is . Dropping elementsfrom the second sum can only increase the overall result,and hence we obtain

(71)

Combining (59), (60), and (71), we obtain

(72)Combining (72) with (70), recalling (21), we obtain ourdesired result of (20).

F. Proof of Theorem 6

The proof of this theorem follows in lines similar to thoseused in the proofs of Theorems 2 and 3 of [21].

Using a union bound, we obtain

(73)

where and are determined later. We now proceed to boundboth elements of the above sum.

1) Requiring and invoking (19) we obtain

(74)

As in the proof of Theorem 2 in [21], we define

Using methods similar to those in [21] we obtain

Recalling and we obtain

We now define

The inner contents of the braces are positive for all andapproach as . Therefore, the above value ispositive. Recalling and we have

We now return to (74) and obtain

(75)

2) We now examine values of satisfying .From (20) we have

(76)

We now restrict , and obtain, recalling (21) andusing

Using for all , we have

The function ascends from zero inthe range . Defining

we obtain that for all in the range

We now define as some value in the rangeyielding . Using (76) we obtain

Therefore,

(77)

Combining (73) with (75) and (77) we obtain our desired resultof (22).


G. Proof of Theorem 7

Let be defined as in Theorem 6. Let be some numbersmaller than that will be determined later. Let be the un-derlying expurgated LDPC ensemble of , obtained by expur-gating codes with minimum distance less than or equal to .We use Theorem 3 to bound .

Assigning

(78)

we have (using (12))

(79)

We now examine both elements of the above sum.

1) As in the proof of Theorem 2, we obtain that for all. We also obtain that for each ,

cannot contain more than one codeword having .Examining for we have

Letting denote the spectrum of code , wehave

Summing over all and taking the average spectrum weobtain

Finally, selecting so that we obtain

(80)

2) can be written as

where is defined as in (17). We now examine

(81)

Examining (23) it is easy to verify that the first elementof the above sum approaches zero as . To boundthe second element, we first rely on (47), (68), and (69),and obtain a finite- bound on

Thus, the second element in (81), evaluated over all valid, is bounded arbitrarily close to zero as . The

third element is upper-bounded using Theorem 5, for ,satisfying and . The fourth elementapproaches zero as by (68) (recalling

).Summarizing, there exist positive integers , such

that for , , satisfying , , andthe following inequality holds:

(82)

Combining (79), (80), and (82) we obtain our desired result of(24).

H. Restriction to Prime Values of

Let be a positive integer, and let be a nonprimenumber. We now show that there exist values of such that no

exists as in Theorem 5. Moreover, expurgation is impossiblefor those values of .

To show this, we examine the subgroup of the modulo-group, formed by . This subgroup is isomorphic to thegroup modulo- (the binary field). Given a modulo- LDPCcode , we denote by the subcode produced by codewordscontaining symbols from alone. We now examine theensemble of subcodes .

The binary asymptotic normalized spectrum is defined, foras

The limiting properties of this spectrum are well known (see[17]). Fixing the ratio and letting ,

approaches the value where

is the binary entropy function.


We now examine values of belonging to the set definedby

Given the above discussion, the normalized spectrum ofmodulo- LDPC, defined by (14), evaluated over , corre-sponds to . We therefore obtain that for

We now show that expurgation is impossible. The probabilityof the existence of words of normalized type in , in a ran-domly selected code, corresponds to the probability that any bi-nary word satisfies the constraints imposed by the LDPC parity-check matrix. This results from the above discussed isomor-phism between and the modulo- group. This probabilityclearly does not approach zero (in fact, it is one), and, hence,expurgation of codes containing such words is impossible.

APPENDIX IIIPROOFS FOR SECTION IV

A. Proof of Theorem 8

The proof of this theorem follows in the lines of the proof ofTheorem 4.

Equation (48) is carried over from the modulo- case. Letbe a word of type . Recalling Lemma 2, we can assume

without loss of generality that is a binary word of weight.

Each assignment of edges and labels infers a symbol “col-oring” on each of the check node sockets (the th color corre-sponding to the th element of GF ), according to the adjacentvariable node’s value multiplied by the label of the connectingedge. Exactly of the sockets are assigned colors of the set

. All color assignments satisfying the above re-quirement are equally probable. The total number of coloringsis

(83)

Given an assignment of graph edges, is a codeword if at eachcheck node, the GF sum of the values of its sockets is zero.The number of valid assignments is determined using the enu-merating function

(84)

where is the weight-enumerating function of -lengthwords over GF satisfying the requirement that the sum ofall symbols must equal over GF .

Using (48), (51) (with , and replaced byand ), (83) and (84), and applying Theorem 10

as in the proof of Theorem 4, we arrive at (25). We now showthat is given by (26).

We first examine , defined such that the co-efficient of is the number of words of type

(the index corresponds to the th element ofGF ) whose sum over GF is zero. A useful expression for

is

(85)Elements of GF can be modeled as -dimensional vectorsover (see [3]). The sum of two GF elementscorresponds to the sum of the corresponding vectors, evaluatedas the modulo- sum of each of the vector components. Thus,adding elements over GF is equivalent to adding -di-mensional vectors. Letting

we have the following expression for :

Consider, for example, the simple case of and .To simplify our notations, we denote by and by

where and are evaluated modulo- . This last equa-tion is clearly the output of the two-dimensional cyclic convo-lution of with itself, evaluated at zero. In the general case(85), we have the -dimensional cyclic convolution of the func-tion with itself times, evaluated at zero.Using the -dimensional DFT of size to evaluate the convo-lution, we obtain

IDFT DFT

We now examine in (86) at the top ofthe following page. Consider the elements


(86)

These elements correspond to the DFT of the function, which is a delta function. Thus, we obtain

Combining the above with (86) we obtain our desired resultof (26).

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments.

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