Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 9, SEPTEMBER 2013 5905

Quasi-Perfect Codes From Cayley GraphsOver Integer Rings

Cátia Quilles Queiroz, Cristóbal Camarero, Carmen Martínez, and Reginaldo Palazzo, Jr., Senior Member, IEEE

Abstract—The problem of searching for perfect codes has at-tracted great attention since the paper by Golomb and Welch, inwhich the existence of these codes over Lee metric spaces was con-sidered. Since perfect codes are not very common, the problem ofsearching for quasi-perfect codes is also of great interest. In thisaspect, also quasi-perfect Lee codes have been considered for 2-Dand 3-D Lee metric spaces. In this paper, constructive methodsfor obtaining quasi-perfect codes over metric spaces modeled bymeans of Gaussian and Eisenstein–Jacobi integers are given. Theobtained codes form ideals of the integer ring thus preserving theproperty of being geometrically uniform codes. Moreover, they areable to correct more error patterns than the perfect codes whichmay properly be used in asymmetric channels. Therefore, the re-sults in this paper complement the constructions of perfect codespreviously done for the same integer rings. Finally, decoding algo-rithms for the quasi-perfect codes obtained in this paper are pro-vided and the relationship of the codes and the Lee metric ones isinvestigated.

Index Terms—Cayley graphs, Eisenstein–Jacobi (EJ) integerrings, Gaussian integer rings, geometrically uniform codes,quasi-perfect codes.

I. INTRODUCTION

G eometrically uniform codes were proposed by Forney[8]. This class of codes encompasses the Slepian Group

codes and the Lattice codes by allowing the elements of the gen-erator group to be arbitrary isometries of the Euclidean spaceinstead of orthogonal transformations or translations when con-sidering the previous classes separately. A space signal code isdefined as geometrically uniform if for any two code sequences,there exists an isometry that takes a code sequence to the otherwhile leaving the code invariant. Such a code has desirable sym-metry properties such as: the Voronoi regions are congruent,the distance profile is the same for any codeword, each code-word has the same error probability, and the generator group is

Manuscript received June 20, 2012; revised February 21, 2013; acceptedApril 24, 2013. Date of publication June 06, 2013; date of current versionAugust 14, 2013. C. Q. Queiroz and R. Palazzo, Jr., were supported bythe FAPESP under Grant 2007/56052-8 and by the CNPq under Grant303059/2010-9. C. Camarero and C. Martínez were supported by the SpanishMinistry of Science under Contracts TIN2010-21291-C02-02, AP2010-4900,and CONSOLIDER Project CSD2007-00050, and by the European HiPEACNetwork of Excellence.C. Q. Queiroz is with the Department of Mathematics, Universidade

Federal de Alfenas, Alfenas, Minas Gerais 37130-000, Brazil (e-mail:[email protected]).C. Martínez and C. Camarero are with the Department of Computers and

Electronics, Universidad de Cantabria, Santander, Cantabria 39005, Spain(e-mail: [email protected]; [email protected]).R. Palazzo, Jr., is with the Department of Telematics, Universidade Estadual

de Campinas, Campinas, Sao Paulo 13083-852, Brazil (e-mail: [email protected]).Communicated by J.-C. Belfiore, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2013.2266398

isomorphic to the permutation group acting transitively on thecodewords.In [9], Golomb and Welch define close-packed codes or per-

fect codes by the use of polyominoes, where each codeword hasa decision region, Voronoi region, given by Lee spheres of ra-dius . These regions satisfy the property that there is a groupacting transitively on them which cover the torus and conse-quently, the resulting code is geometrically uniform. In [6], flattori were used with a similar objective, as well as signal setsof the QAM-type considered as perfect coset codes with the in-duced distance from the Euclidean metric.In [11] and [12], quotients of the Gaussian and Eisenstein–Ja-

cobi (EJ) integer rings were proposed for modeling QAM-typeand hexagonal signal constellations. Later, in [14]–[17] a newmetric over these spaces was introduced. This metric, similarto the Lee metric, is defined by means of the Gaussian and EJgraphs, which are Cayley graphs over the integer rings mod-eling the signal constellation. In [19] and [7], the main dis-tance-related properties of Gaussian and EJ graphs were char-acterized, providing closed expressions for their diameter andaverage distance. On the other hand, perfect codes were con-structed as being ideals of the integer rings, thus solving thetheoretical problem over the graphs known as the perfect dom-inating set of the vertices. The problem of quasi-perfect codesover Gaussian and EJ graphs was previously considered in [20]and [21]. In this paper, a complete characterization of such codesbeing ideals which corrects some of the results in the previousworks is presented.The construction of quasi-perfect group codes was also

considered in [4] for the Lee metric. This kind of codes haveshown to have different practical applications, as in phasemodulated and multilevel quantized-pulse-modulated channels[5], [9]. Moreover, they constitute the solution to the optimalresource allocation in toroidal interconnection networks as itwas considered in [2] and [3] for the 2-D and 3-D cases. Inaddition, decoding algorithms for Lee-distance quasi-perfectcodes were presented in [4] and [10].The aim of this paper is to provide the construction of quasi-

perfect codes over the Gaussian and EJ integers, which besidespreserving the geometrically uniform structure of the codes theyare able to correct more error patterns than the perfect codes.Moreover, since in [17] the relationship between perfect codesfor the 2-D Lee spaces and perfect codes over Gaussian integerswas shown, in this paper the relation between the constructiongiven in this paper with the quasi-perfect Lee codes is also in-vestigated. It will be shown that new quasi-perfect codes over2-D spaces can be obtained from the methods given here. Thenew quasi-perfect codes not only are groups but also form idealsover the integers rings.

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5906 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 9, SEPTEMBER 2013

Finally, decoding algorithms for the presented codes whichtake advantage of the ideal structure of the constructions aregiven. The algorithms which will be considered have some ge-ometrical similarities to the one presented in [10] for the Leemetric although they decode different code constructions.The remaining of the paper is organized as follows.1) In Section II, some basic concepts from number theorywhich are necessary to define quotient rings of Gaussianand EJ integers are presented. Moreover, graphs over quo-tients of these integer rings and codes over them are consid-ered. Also, perfect and quasi-perfect codes over graphs aredefined and geometrically uniform codes are constructedover them.

2) In Section III, an extension of perfect codes is made by pre-senting quasi-perfect codes over quotient rings of Gaussianintegers. In addition, a complete characterization of quasi-perfect codes being ideals of the ring is given.

3) Similarly to the previous section, in Section IV quasi-per-fect ideal codes over quotient rings of EJ integers are char-acterized.

4) In Section V, the relation between the quasi-perfect codespresented in Section III and previously known Lee metricquasi-perfect codes is considered. Necessary and sufficientconditions for a group code being an ideal are given. Asa consequence, the constructive method presented in thispaper gives new quasi-perfect codes over 2-D Lee spaces.

5) In Section VI, some decoding algorithms for the codes pre-sented in this paper are obtained. The decoding algorithmstake advantage of the algebraic structure of the codes, thatis, that they form ideals.

6) Finally, in Section VII some conclusions are drawn.

II. PRELIMINARY RESULTS

This section is organized into three subsections. In the firstone, quotients of Gaussian and EJ rings, which will be used todesign signal constellations, are introduced. In the second sub-section, Cayley graphs over these quotient rings are consideredin order to define metrics over the previous rings. Finally, in thethird subsection, definitions of perfect and quasi-perfect codesover these graphs are stated.

A. Quotient of Integer Rings

Next, basic results from number theory which are importantfor the development of the remaining sections are presented.More detailed information can be found in [22], [23], and [13].Let be a number field with degree and the

monomorphisms of into . For any the norm of isdefined as

Since the s are monomorphisms, it follows that

for any . In addition, if , then .Now, given a number field , let be the ring of integers

of and the ideal of generated by . Then, thefollowing result is obtained.

Proposition 1 ([22, Proposition 1, p. 52]): Let .Then,

that is, the quotient ring has elements.The codes presented in this paper will be constructed

over signal constellations modeled by Gaussian and EJintegers. Therefore, let us first consider the number field

, where . The ring ofintegers of is , called the ring of the Gaussian integersand denoted by . Ifthen its norm is given by

that is, the norm is given by a quadratic form such that

Now, let , called the ring of the EJ integers and denotedby , where . Note thatis such that . Hence, if thenits norm is given by

that is, in this case its norm is given by a quadratic form suchthat

If denotes the ideal of generated by , where eitheror , then the quotient ring generated by such an

ideal is

where . From Proposition 1 it follows that:Theorem 2 ([15], [16]): Let with .

Then, has elements.Moreover, using the third ring isomorphism theorem, [13], it

can be easily inferred the following consequence of the previousresult.Corollary 3([15], [16]): Let with .1) If is such that , then the idealhas order ;

2) If is such that and , thenthe ideal is generated by and has order

.Example 1: Given then its norm is .

Hence, from Theorem 2, has 25 elements, which areobtained from the quotient of the ring by the ideal, or equivalently, by taking modulo . Hence,

,,

.On the other hand, if , its norm is and

the induced quotient ring has 37 elements. In Fig. 1,both quotients are graphically represented.

QUEIROZ et al.: QUASI-PERFECT CODES FROM CAYLEY GRAPHS OVER INTEGER RINGS 5907

Fig. 1. Signal constellations obtained as and .

B. Graphs Over Gaussian and EJ Integer Rings

Since the codes presented in this paper are obtained from quo-tients of Gaussian and EJ integers, metrics over these rings mustbe defined. In this direction, Gaussian and EJ graphs are definedas Cayley graphs over the quotient rings, so the metric inducedby these graphs will be the one considered for the code con-struction. As it is well known, Cayley graphs are defined overgroups as follows.Definition 4: The Cayley graph over a group with adja-

cency set is defined as the graph with vertexset and edge set

Hence, Gaussian and EJ graphs are Cayley graphs where thecorresponding adjacency sets are the units of the integer rings.Definition 5: Let , where .1) If then the Gaussian graph generated by is definedas .

2) If then the EJ graph generated by is defined asEJ .

As it has been remarked in the previous section, the orderof the graphs is given by the norm of its generator. Clearly,Gaussian graphs are regular of degree 4 and EJ-graphs havedegree 6. Since they have been defined as Cayley graphs, theyresult in vertex-symmetric graphs, that is, for any pair of ver-tices there is an automorphism which sends one into the other.As a consequence, the distance distribution of the vertices canbe determined by counting the number of vertices at each dis-tance from a central vertex, which is usually selected to be zero.The complete determination of these distributions has been donein [19] and [7] for Gaussian and EJ-graphs, respectively. As aconsequence, the diameter of the graph, that is, the length ofthe longest shortest path has been exactly determined. Next, theresults summarizing the distance distributions are presented inorder to be self-contained.Theorem 6 ([19]): Let and .

Let and for any positive integer , let be thenumber of vertices in at a distance . Then,

ififififif

if and modif

Fig. 2. Graphs and EJ .

Theorem 7 ([7]): Let and .Let and . For any positive integer , let

be the number of vertices in EJ at a distance . Then,

ififif

if mod andif

Example 2: In Example 1, the set of vertices isand the set of edges as in the previous definition, is shown inFig. 2. The diameter of the graph is 3 since every vertex is atdistance less than or equal to 3 from the central vertex. More-over, as it can be seen this graph has the maximum number ofvertices for diameter 3, or equivalently, it is dense.1 On the otherhand, the graph EJ has the set of verticesand the adjacency is completed as shown in Fig. 2. In this case,the diameter of the graph is also 3, and the graph is also dense.Now, to define a metric over the integer rings considered in

this paper it is only needed to consider the distance induced byits corresponding Gaussian or EJ graph.Definition 8 ([15], [18]): Let , where. The distance in is the distance induced by the as-

sociated Cayley graph or EJ . Thus, if , thegraph distance can be computed as:1) , such that

, where .2) , such that

, where .Remark 9: Note that the distance between any two vertices is

the length of any shortest path between them. Then, the diameterof the graph gives the maximum distance in the metric space.As a consequence, signal constellations corresponding to densegraphs contain the maximum number of signal points for a givenmaximum distance.

C. Geometrically Uniform Codes Over Quotient Rings

Once the metric spaces to be considered for quasi-perfectcodes constructions have been established, classical definitionsof codes will be provided in this section. Given a graph

and distance , a code in is a nonempty subset of. The Voronoi region associated with is the subset of

the elements in for which is the closest point in , that is,

1Note that this is a different definition of dense graph from the usual one,which refers to the number of edges instead of the number of vertices.


From this, the covering radius of the code is defined as

Let denote the ball of radiuscentered at . Then, the covering radius is the least numbersuch that the balls of radius centered at the points of cover. Then,

is the minimum distance of , with . The equalityholds, that is , when the balls of radius centeredat the points of partition . A code satisfying this propertyis called perfect and it is said to correct errors. Next, perfectand quasi-perfect codes, which are the target of this paper, aredefined.Definition 10: Let be a graph and denote its

distance. Let .1) is a quasi-perfect if

a) For every pair of different codewords itfollows that

b) For every vertex , there exists such that.

2) is a perfect if for every vertex thereexists a unique codeword such that .

Perfect codes being ideals for the Gaussian and EJ graphswere obtained in [14] and [15]. These codes have the propertyof being generated by elements with maximum norm for a givendiameter, or equivalently, as codes associated with dense graphs.The following result summarizes how to obtain perfect codesover Gaussian and EJ-graphs.Theorem 11 ([15], [18]): Let ,

and be a positive integer.1) If and (or ) divides , then theideal (resp. ) is a perfect over .

2) If and (or ) divides , then theideal (resp. ) is a perfect over EJ .

Note that since the construction is made by means of idealsof the integer ring, the resulting codes are not only perfect butalso geometrically uniform. In fact, it is straightforward that anyideal over the quotients generates a code over the graph, as it isproved in the following result.Corollary 12: Let , . Then, ifdivides , then the ideal forms a geometrically uniform

code over . Moreover,1) If then the code can correct every error pattern ofweight , for .

2) If then the code can correct every error pattern ofweight , for .Proof: The ideal generates a geometrically uniform code

straightforwardly. On the other hand, the error capacity is ob-tained as a consequence of Theorems 6 and 7.Example 3: Given then . Hence,

from Theorem 2, has 545 elements. Now,may be written as . There-

fore, generates a geometrically uniform code over

Fig. 3. Geometrically uniform code generated by over .

which corrects every error pattern of weight . More-over, the distance distribution of the graph can be directlyinferred from Theorem 61)2)3)4)5)6)7)8)9)10)Therefore, the code generated by corrects 12 error patterns

with errors, 8 with , and 4 with ,resulting in the 24 errors. This geometrically uniform code isshown in Fig. 3. As it can be checked, the code is not a perfectcode neither a quasi-perfect code.In the next sections the problem of characterizing quasi-

perfect codes over Gaussian and EJ-graphs is considered, thuscomplementing the previous works on perfect codes overGaussian and EJ-graphs. The codes considered will also begenerated as an ideal, thus obtaining geometrically uniformcodes. Also, being ideals will simplify the decoding procedures.Finally, by using Theorems 6 and 7 the distance distribution ofthe codewords can be calculated as done in the previous example.

III. QUASI-PERFECT CODES OVER QUOTIENT RINGS OFGAUSSIAN INTEGERS

In this section, a constructive method for obtaining quasi-perfect codes over Gaussian integer rings is presented. As ithas been mentioned before, given the signal constellationequipped with the Gaussian metric, it is enough to choose adivisor of , such that is either of the form or

, to obtain a perfect code. The code is defined asthe ideal generated by the divisor and the signal constellation iscovered by the fundamental region whose covering radius hasmaximum value of . That is, the fundamental regions consist


of Lee spheres of radius . Thus, perfect codes are obtained bytranslations of Lee spheres of radius . As observed in [9], a Leesphere with radius has cells. By Corollary 12, anybeing a divisor of , defines a new code, although not neces-

sarily a perfect code, but with the property of covering the signalconstellation by identical fundamental regions. In this section,the characterization of a divisor such that the ideal generates aquasi-perfect code is done. In this aim, some distance propertiesof Gaussian graphs will be needed. In Theorem 6, the verticesdistance distribution has been presented and as consequence, itcan be obtained in the following result.Corollary 13: Let and consider . Then,1) The value gives the radius of the max-imum Lee sphere contained in the Voronoi region associ-ated with .

2) If is odd, the value is the max-imum distance from any word to the center of the Voronoiregion. If is even, the value isthe maximum distance from any word to the center of theVoronoi region.Proof: This corollary is a consequence of Theorem 6.

As a consequence and considering [20], a constructivemethod for quasi-perfect codes that gives a complete character-ization can be obtained as presented in Theorem 14.Theorem 14: Let and be a positive integer. Let

. Then,1) If divides , then the ideal forms a quasi-perfect

-code over .2) If divides , then the ideal forms a quasi-perfect

-code over .In both cases, the code can correct every error pattern of

weight and error patterns with weight. Moreover, these are the unique ideals which form quasi-per-fect codes over .

Proof: Let us consider the first item since the other one canbe demonstrated in a similar way. Now, for every

it is of the form orwith integers ,

depending on the parity of its norm. It is enough to considerthe values and for with odd norm and

and for with even norm. Hence, we canassume , which implies .First, we consider that

divides and define . Now, given twodifferent codewords, it has to be proved that isan empty set. Clearly, this can be obtained from the first itemof Corollary 13 since gives the radiusof the maximum Lee sphere contained in the Voronoi regionassociated with .Now, let us consider . Then, by Corollary 13 it is ob-

tained that is themaximum distance from any word to the center of the Voronoiregion. Therefore, for any there must exist a codeword

such that . Now, if and onlyif , thus obtaining the first two values for in thetheorem. Moreover, gives us the perfect -code.Second, let us assume that

. The proof is similar to the previous case,

Fig. 4. Tiles of the three quasi-perfect 3-codes over .

however for the even cases of Gaussian generators. Note thatgives the radius of the maximum

Lee sphere contained in the Voronoi region associated with .Now, sinceit follows that this is the maximum distance from any word tothe center of the Voronoi region. Then, clearly if andonly if .To conclude with the proof just note that the norm of is

cardinal of the Voronoi region generated by andis the number of vertices at a distance less than or equal to .The distance distribution of these three quasi-perfect codes is

shown in Fig. 4 for error correction capacity . Note thatthe one in the middle corresponds with the perfect code and thatvertex zero is highlighted in white.Example 4: Let us consider , which implies

. Hence, for any multiple of a Gaussianring with a quasi-perfect 3-code is obtained. Let us considerfor example . Then, theideal forms a quasi-perfect 3-code with

codewords. Now, if we consideras a multiple of the previous generator, that is,

, then the ideal forms againa quasi-perfect 3-code however withcodewords. A graphical representation of both sets over their

corresponding Gaussian graphs is shown in Fig. 5.Remark 15: Note that the uniqueness is strongly obtained

from the condition of being an ideal. If we relax this condition,codes which are not obtained from the construction given inTheorem 14 can be more or less straightforwardly constructed.In Example 5, we show one of these possible codes. A specialcase of quasi-perfect codes being groups, but not ideals will bediscussed in Section V.Example 5: Let us consider . The metric

space induced by the corresponding Gaussian graph haselements. Let us consider the subset

formed by the 46 elements depicted in Fig. 6 as bolded points.It can be checked that this subset forms a quasi-perfect codeover but it neither forms an ideal nor a group of the ring.Moreover, the obtained code is not geometrically uniform. Notethat is isomorphic to the 2-D Lee space . Moredetails about the relationship between quasi-perfect codes over2-D Lee spaces and the ones presented in this section will bediscussed in Section V.

IV. QUASI-PERFECT CODES OVER EJ INTEGER RINGS

In this section, the characterization of quasi-perfect codesover EJ-graphs generated by ideals is considered. Analogouslyto the Gaussian case, perfect codes over EJ-graphs can be ob-tained over EJ integers modulo by choosing a divisor of


Fig. 5. Quasi-perfect 3-code generated by over and .

Fig. 6. Nongeometrically uniform quasi-perfect code over .

of the form or its conjugate, as it was shown in[18]. Then, the ideal generated by such a divisor forms a perfectcode and the signal constellation is covered by fundamental re-gions whose covering radius has maximum value of , that is, thefundamental regions are hexagons with radius . Thus, perfectcodes are obtained by translations of the hexagons with radiuswhich have cells. Therefore, if a different divisor isconsidered, the generated code is not perfect anymore. In The-orem 17, the adequate values for a divisor such that it generatesa quasi-perfect code are stated. Moreover, it is also shown thatthe divisors provided are the only ones if the wanted code hasto be an ideal.In order to construct such codes, some distance properties of

EJ-graphs should be considered first. The following result is aconsequence of the distance distribution of EJ-graphs given in[7] and summarized in Theorem 7.Corollary 16: Let and consider EJ .

Then,1) The value gives the radius of themaximumLee sphere contained in the Voronoi region associated withEJ .

2) The value is the max-imum distance from any word to the center of the Voronoiregion.

Considering [21] we get the next result that gives a completecharacterization of ideals over EJ-graphs that form quasi-perfectcodes.Theorem 17: Let and let be a positive integer. Let

be such that and,

. Then, the idealis a quasi-perfect -code over . Moreover, these are theonly (up to units multiplication and conjugation) quasi-perfect-codes being an ideal and generates a perfectcode.

Proof: Let us consider such that ,and

, . Then, by Corollary 16 we have to searchfor such that

Now, two cases are considered separately:1)2) .For the first case, it follows, for some integers ,

, that

As a consequence which gives thefollowing possible values for :


Fig. 7. Tiles of the seven quasi-perfect 3-codes over .

Then, considering the different cases and taking into accountthat , the only possibility is for. Note that this value gives the perfect code.Now, for the second case, it follows that

which gives the following values:

Then, considering the different cases and taking into accountthat the remaining of the given divisors areobtained.The distance distributions of the seven quasi-perfect codes

obtained in Theorem 17 are shown in Fig. 7 for error correctioncapacity . Note that the tile situated on the left uppercorner corresponds with the perfect code and that vertex zerois highlighted in white.Example 6: Let us consider and

.This EJ-integer generates a hexagonal signal constellation with

points. Now, if the code isdefined using the first divisor of , that is, , thenthis ideal constitutes a perfect 2-code with 31 codewords. On theother hand, if the code is defined using the other divisor, that is

, then the ideal is in this case a quasi-perfect 2-codewith 19 codewords. Both codes correct all the error patternsfor , but the latter code also corrects error patterns for

.Remark 18: Again, note that the uniqueness of such codes is

conditioned by the restriction of being ideals. Therefore, codeswhich are neither ideals nor groups of the EJ-integers can beeasily constructed as it was done for the Gaussian integers case.Example 7: In Fig. 8, the code generated as a groupover the graph is represented. As it can be checked,

this code is not an ideal since but.

V. QUASI-PERFECT CODES FOR THE LEE METRIC

Quasi-perfect Lee distance codes over were consideredin [4]. Given positive integers and , the proposed code in

is the one generated by the matrix that is, thecode is defined as a group over as follows:

Fig. 8. Group quasi-perfect code over .

In [4], AlBdaiwi and Bose established the conditions oversuch that is a quasi-perfect -code in . Moreover, two dif-ferent decoding schemes are provided in the same paper. Later,in [10] an optimized decoding algorithm is presented for thesame family of codes. In this section, we will refer to this con-struction as the quasi-perfect group code construction.As it was proved in [17], certain perfect Lee codes over 2-D

spaces can be obtained as subcases of perfect codes being idealsover the Gaussian integers. The main idea under this result isthat and are isomorphic as groups such that the twocorresponding metrics coincide. That is, the Lee distance andthe one induced by theGaussian graph are the samemetric, sincethe underlying graphs are isomorphic. Therefore, Theorem 14can be applied also to obtain quasi-perfect Lee codes over .Moreover, it makes sense to consider the relationship betweenthe two families of codes, that is, the one given by the quasi-perfect group codes construction and the one presented in thispaper.As a first approach to the determination of the connection be-

tween the two types of codes, note that the codes constructedusing Gaussian integers are indeed ideals over the Gaussian in-teger rings, while the codes defined by AlBdaiwi and Bose in [4]are just group codes over the Gaussian integers. Therefore, inthe general case, both codes do not coincide. As an example, letus consider the Lee space . Clearly, this space can be seen as

with the Gaussian graph’s metric. In Example 8, two dif-ferent constructions of quasi-perfect Lee codes over this spaceare considered, the first one being a group and the second onebeing an ideal, both over the same Gaussian integer ring.Example 8: Let us consider . If we change the

notation to Gaussian integers, in [10] it was shown that the codegiven by the group is a quasi-perfect3-code, as shown in Fig. 9. Note that the code has 29 codewords.On the other hand, we can consider by Theorem 14 the ideal

over . Again, this code is a quasi-perfect3-code with 29 codewords, since .Now, if the distance properties of the codes are considered,

it can be seen that both codes have maximum distance 25 andaverage distance 15. However, the codes are different and let us


Fig. 9. Quasi-perfect code over being a group but not an ideal.

Fig. 10. Quasi-perfect code over being an ideal.

illustrate it. In the first group code , for every codeword thereexist two codewords at distance 7 and two codewords at distance8 from it. On the other hand, in the code being an ideal , forevery codeword there are exactly four codewords at distance 7.The next codewords are obtained at distance 10, as it can be seenin Fig. 10.What comes out from the previous example is that both con-

structions are different in general. Moreover, the question ofwhen both the construction of quasi-perfect group codes andthe one presented in this paper coincide is directly connectedwith the study of the situation in which both algebraic structuresmatch up, that is, whenever an additive group of the Gaussianintegers is also an ideal over the same ring. In this direction, weconsider the following lemma which characterizes these situa-tions. Let denote the greatest common divisor over theGaussian integers and the greatest common divisor overthe ring of integers.

Lemma 19: Let and consider. Let Then, if and

only if .Proof: Let us denote ,

and .First, it will be proved that if then .

Since it follows that for someinteger . Thus, there exists such thatand with . Since divides

and and are coprimes, it follows that divides. This entails that divides

As a consequence, the following Diophantine system isobtained:

for integers . Now,

By simplification, it is obtained that which implies.

To prove the converse, let us assume , thatis, there exist integers such that . Let be

. Then,. As and

it follows that . Hence,and , which concludes the proof.As a consequence of the previous lemma, it is expected that

both constructions, although different in the general case, yieldthe same codes in certain cases. In the following example, sucha situation is considered, that is, we construct a quasi-perfectLee group code which results in an ideal code over the Gaussianintegers. However, as it can be seen in that example, the samequasi-perfect code cannot be obtained from both constructionsseparately.Example 9: Let us consider

and the ring . In this ring, a quasi-perfect code whichis a group and also an ideal is being constructed. Therefore,the example illustrates that the previously known constructiongiven in [4] and the one presented in this paper are not com-pletely disjoint in the quasi-perfect case. Hence, let us consider

the generator of both codes.Note that , with and

, fulfilling the hypothesis of the previous Lemma19. Now, it is enough to realize that

where which gives a 6-quasi-perfectideal code using Theorem 14, with and .


Remark 20: Note that in the previous example, the groupcode is generated by an element of the form but itscorrection capacity is not equal to .Moreover, it can be straight-forwardly obtained that the only group codes from the construc-tion in [4] that coincide with the ideal quasi-perfect codes con-sidered in this paper are in fact the perfect codes generated by

. As a consequence, the generators given inTheorem 14 provides many new examples of quasi-perfect Leecodes over 2-D spaces.

VI. DECODING ALGORITHMS

In this section, decoding algorithms for the quasi-perfectcodes over Gaussian and EJ-integers obtained in this paperare presented. The algorithms take advantage of the algebraicstructure of the codes. Hence, the procedures use the fact thatthe codes form ideals over the corresponding integer ring toperform the decoding process.Decoding algorithms for the Lee-distance quasi-perfect codes

were presented in [4] and [10]. AlBdaiwi and Bose’s algorithmin [4] makes a strong use of the cyclic nature of the group codesthat they consider. They construct a subset of the code with car-dinality and correct by the closest codeword. Therefore,the algorithm can be straightforwardly adapted to decode ourcodes when these codes are additive cyclic groups over the in-teger rings, as it has been considered in Lemma 19. A more ef-ficient algorithm also for decoding quasi-perfect Lee codes waspresented by Horak and AlBdaiwi in [10]. In this case, the re-ceived symbol is corrected by the closest codeword among atmost four codewords. The algorithms that are proposed in thissection have some geometrical similarities to this last one al-though they decode different code constructions. Finally, in [1]and [7] general algorithms for minimum distance calculationsare given, which can be used for decoding in Gaussian and EJlattice constellations. However, in this paper we give a differentapproach which tries to minimize integer multiplications anddivisions.The decoding procedures, both for the Gaussian and

the EJ-integer rings, are presented in Algorithms I and II,respectively. However, both methods follow the same idea.Since the codes are defined by means of ideals, the codewordsare multiples of the generator of the ideal. Hence, given areceived word, finding the nearest codeword which correctsit is equivalent to finding the quotient which results from theEuclidean division of the received word by the ideal generator.Moreover, the correctness of the algorithms is guaranteedby Theorem 28, which is obtained as a consequence of thefollowing lemmas.Notation 21: As it can be seen in the algorithms’ descrip-

tion, will denote the common Manhattan weight for a givenGaussian or EJ-integer. Let us also denote by the roundingoperator, with . Then, the quotient and

the remainder will be denoted as and

It can be checked thatwith , which

provides a Euclidean division algorithm for ,Lemma 22: Let . If

then .

Proof: Let , . Then, it follows that

Algorithm I Decoding in

Data being the generator

being the code generator

being the code correction capacity

being the received symbol

Result being the corrected symbol

Compute , ;

if or then

Return

else

Compute:

Find such that

Return

end

Algorithm II Decoding in

Data being the generator

being the code generator

being the code correction capacity

being the received symbol

Result being the corrected symbol

Compute , ;

if then

Return

else

Compute:

Find such that

Return

end

The next result considers the geometrical location of the re-mainders obtained by such a division algorithm.


Lemma 23: Let . The set

is obtained as the integral points of the complex parallelepipedwith vertices at , where and

.Proof: Let us define and .

By Lemma 22, it is obtained that. Hence,

and . Since andare both linear functions it follows that is a convex

polyhedron.One pair of vertices of is with. Thus, , from

which it is obtained that .The other pair of vertices of is with

. Thus,, from which it is obtained that .

As a consequence, the remainders generated by are lo-cated in a parallelepiped . A translation of this parallelepipedwas considered in [7] for defining a set of representatives ofthe quotient group. Moreover, the weight of the remainder ob-tained over the Gaussian integers is bounded by the diameterof the Gaussian graph generated by the divisor. However, thisremainder not always minimizes the weight as it is expected inorder to perform correction. The next lemma proves these facts.Lemma 24: Let .

if modif mod

Proof: Let and be as in Lemma 23. Since , thevertices of are . Now, as is a linear function,it has to be maximized in a vertex. Thus,

. If 2 does not divide ,cannot be and as a consequence the strict

inequality is obtained.Since the remainder considered in the previous lemma not al-

ways minimizes the weight function, slight modifications mightbe done in order to find the one with minimum weight, as it isshown in the following result.Lemma 25: Let be a quasi-perfect -code over

such that is odd. Then, for every , there existssuch that.

Proof: Let be the closest codeword to . Ifthen let . Otherwise, as is the closest codeword it

follows that . By Lemma 24 and the fact thatnecessarily . Now,

by the triangular inequality. As and is a quasi-perfect -code, then

. Finally, .Lemma 26: Let be a quasi-perfect -code over such

that is even. Then, for every ,.

Proof: By uniqueness, it can be assumed that. For any let

. If , then by Lemma 24 it follows

that with minimum weight. Otherwise,. Then,

As a consequence,

and . Hence,

and .Although in the EJ integers case the corresponding remainder

may have a larger weight, the operations to get theminimum oneare similar, as it is shown in the next result.Lemma 27: Let and be a quasi-perfect-code over . For every , there existssuch that .

Proof: Let us consider , , and as in Theorem 23. Itis clear that the result only has to be proved for the boundaryof . Then, is the middle point between and

and is the middle point between 0 and.

Let us prove first that all the points in the segment fromto can be corrected by 0 or . In this direction, note thatcan be corrected by . Since the other points in the segmentare farther away from , no point in the segment is correctedby . Similarly, can be corrected by 0 and the other points arefarther away from .By an analogous reasoning regarding the other segments it

follows that the only codewords which correct points in are.

Theorem 28: Algorithms I and II are correct.Proof: Lemmas 25 and 26 guarantee the correctness for

with odd and even norms, respectively. Lemma 27guarantees the correctness of the algorithm over .Remark 29: Although the algorithms have similar appear-

ance for Gaussian and Eisenstein integers, the case whereover the Gaussian integers is an exceptional case (indeed isat most ). However, in the EJ case this is a usual situation.Moreover, it can be computed that .Finally, the next example shows the performance of Algo-

rithm II over a particular scenario.Example 10: Let us consider and

. Since and forthen is a quasi-perfect -code over for

error correction capacity . Now, let us assume the symbolhas been received. First, is not a codeword. Then,

let us correct using Algorithm II. By division, we have that

Since then it can be corrected byNow, if the received symbol is , we can proceed

in a similar way. By Algorithm II, we obtain that

with . The set. Note that


Fig. 11. Quasi-perfect 2-code over .

So both and correct the obtainedsymbol. In Fig. 11, a graphical representation of the constella-tion is shown.

VII. CONCLUSION

QAM-type and hexagonal signal constellation have been pre-viously modeled by means of quotients of Gaussian and EJ-in-tegers, [11], [12]. Cayley graphs over these rings were proposedin [15] and [18] to define the so-called Gaussian and EJ metricsover these spaces. Moreover, the problem of perfect codes overthese quotient rings has been previously considered and suchperfect codes were built as ideals over the rings generated byelement with maximal norm in the ring [14].In this paper, quasi-perfect codes over Gaussian and

EJ-graphs have been considered. Constructive methods forquasi-perfect codes being ideals have been given and theuniqueness of the codes, under the hypothesis of being ideals,has been proved. As a consequence, previously known perfectcodes are shown to be the unique ones being ideals over thesegraphs. Moreover, decoding algorithms for the quasi-perfectcodes over Gaussian and EJ-integers have been presented,which also decode the previously known perfect codes.The relationship between perfect codes over Gaussian graphs

and the perfect codes for the 2-D Lee space was considered in[17]. As it was shown, some quotient rings of the Gaussian in-tegers and the 2-D Lee space coincide. Thus, the quasi-perfectcodes construction given in this paper can be also applied togenerate new quasi-perfect codes over Lee spaces. Moreover,the connections between quasi-perfect codes and the previouslyknown for the Lee metric [4] have been investigated. It has beenshown that both constructions are different in the general caseby the characterization of the conditions under which both con-structions collapse.Finally, it can be guessed that the procedures used in this

paper may be extended to other rings, resulting in the construc-tion of new quasi-perfect codes associated with different signalconstellations.

REFERENCES

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[13] T. W. Hungerford, Algebra. New York, NY: Springer-Verlag, 1974.[14] C. Martínez, R. Beivide, and E. Gabidulin, “On the perfect t-domi-

nating set problem in circulant graphs and codes over Gaussian inte-gers,” in Proc. IEEE Int. Symp. Inf. Theory, Adelaide, Australia, Sep.2005, pp. 254–258.

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[21] C. Quilles Queiroz and R. Palazzo, Jr, “Geometrically uniform quasi-perfect codes derived from graphs over integer rings,” in Proc. 3rdInt. Castle Meet. Coding Theory Appl., Cardona, Spain, Sep. 2011, pp.239–244.

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CátiaQuilles Queiroz received the Ph.D. degree in Electrical Engineering fromthe State University of Campinas in 2011. She is a Professor in the Institute ofExact Sciences, Federal University of Alfenas, Brazil. Her research interestsinclude graph theory with applications to coding theory, and design of commu-nication systems in non-Euclidean spaces.


Cristóbal Camarero received the Master’s Degree in Computer Science (withdistinction) from the University of Cantabria, Spain, in 2011. He is currentlyworking towards the Ph.D. degree at the Electronics and Computers Departmentfrom the University of Cantabria. His research interests include graph theorywith applications to interconnection networks and coding theory.

Carmen Martínez received the Ph.D. degree in Mathematics from the Univer-sity of Cantabria (Spain) in 2007. Currently, she is an Associate Professor in theDepartment of Electronics and Computers of the University of Cantabria. Herresearch interests include graph theory with application to coding theory andoptimal topologies for interconnection networks.

Reginaldo Palazzo, Jr., received the Ph.D. degree in Electrical Engineeringfrom UCLA in 1984. He is a Professor in the School of Electrical and Com-puter Engineering, State University of Campinas, Brazil. His research interestsinclude classical and quantum coding theory, genomic coding and design ofcommunication systems in non-Euclidean spaces.

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Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

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