classification of binary constant weight codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010 3779

Classification of Binary Constant Weight CodesPatric R. J. Östergård

Abstract—A binary code � ��

� with minimum distance atleast � and codewords of Hamming weight � is called an �� constant weight code. The maximum size of an �� constantweight code is denoted by�� , and codes of this size are saidto be optimal. In a computer-aided approach, optimal �� constant weight codes are here classified up to equivalence for � �� (with one exception); � � �� ; and � � �� . Moreover, several new upperbounds on �� are obtained, leading among other things tothe exact values �� , and �� .Since �� , this gives the first known example of pa-rameters for which �� with � � ��.A scheme based on double counting is developed for validating theclassification results.

Index Terms—Classification, code equivalence, constant weightcode, double counting, Johnson bound.

I. INTRODUCTION

A BINARY CODE of length is a nonempty set ,where is the field of order 2. The (Ham-

ming) distance between two words is the numberof coordinates in which they differ, and the (Hamming) weight

of a word is the number of nonzero coordinates. Con-sequently, and , whereis the all-zero word. The support of a word is the set of nonzerocoordinates. A code with all codewords of the same weightis a constant weight code. Throughout the paper, all codes areassumed to be nontrivial in the sense that .

The minimum distance of a code is the minimum Hammingdistance between any two distinct codewords. A code withlength , minimum distance greater than or equal to , andconstant weight is called an constant weight code.A constant weight code has even minimum distance , so it isassumed in the sequel that is even. The maximum possiblenumber of codewords in an constant weight code isdenoted by , and the corresponding codes are calledoptimal. The determination of is one of the centralproblems in coding theory, and tables of bounds and exactvalues for this function have been published along the years[1], [6], [10], [14], [25], [40]. The main effort has been put onthe lengths , but bounds for longer codes have also beenpublished [40].

Manuscript received January 13, 2010; revised March 17, 2010; currentversion published July 14, 2010. This work was supported by the Academyof Finland by Grant numbers 110196, 130142, and 132122.

The author is with the Department of Communications and Networking,Aalto University, 00076 Aalto, Finland (e-mail: [email protected]).

Communicated by N. Kashyap, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2010.2050922

Constant weight codes have also been considered in theframework of design theory. A - packing (resp.design) consists of a -set of points and a collection of

-subsets of , called blocks, such that each -subset ofoccurs in at most (resp. exactly) blocks. The largest size of a- packing is given by the packing number ,

where may be omitted in the case .The supports of the codewords of an constant weight

code form a - packing (and vice versa),whereby . Surveys withtables of packing numbers can be found in [31], [42]. Anotherconnection between designs and constant weight codes will beconsidered in Section II.

If one is able to determine the value of for givenparameters, one may try to go one step further and classify thecorresponding optimal codes up to equivalence. This is the topicof the current work and is also stated as [22, Open Problem7.19].

Two constant weight binary codes are equivalent if there is apermutation of the coordinates that maps the codewords of onecode onto the codewords of the other. Such a mapping from acode onto itself is an automorphism, and the set of all automor-phisms of a code forms the automorphism group of the code.

Two general methods for constructing constant weight codesin a classification are discussed in Section II. The isomorph re-jection part of the classification is considered in Section III, andin Section IV it is shown how data from the computations canbe used to validate the results obtained.

The numerical results obtained in this work are tabulated inSection V; the parameters for which codes are classified are

(with one exception);; and . Moreover, several

new upper bounds on are obtained, leading amongother things to the exact values

,and . Since , this gives thefirst known example of parameters for which

with .

II. CONSTRUCTION OF CONSTANT WEIGHT CODES

A brief overview of the problem of classifying constantweight codes can be found in [22, Sec. 7.1.3]. Two main tasksof the classification procedure can be identified: constructionof codes and removal of equivalent specimens of codes. Theformer part will be discussed in this section and the latter part,known as isomorph rejection, in Section III.

There are two obvious ways of constructing constant weightcodes in an incremental manner: codeword by codeword andvia subcodes. As we shall see, both of these are useful and themethod of choice is guided by the main parameters of the code.

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3780 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010

In the more widely applicable method for classifying con-stant weight codes, a code is constructed via one of its subcodes.Given an constant weight code with codewords, itfollows by a direct double counting argument that there mustbe a coordinate with at least 1s. If one shortens thecode by deleting such a coordinate and maintaining only thecodewords with a 1 in the deleted coordinate, then the new codeis an constant weight code. So to obtain an

constant weight code of size , it suffices to con-sider the constant weight codes of size at least

. This step can be iterated to shorter and shorter codes.Analogously, the original code has a coordinate with at least

0s, and one can shorten the code by removingsuch a coordinate and considering the codewords with a 0 in theremoved coordinate. When shortening the code in this way, theweight remains unchanged.

Johnson [19] originally presented these ideas in the form ofthe following bounds.

Theorem 1:

The choice between the two possible types of subcodes inone particular step may have a crucial impact on the speed of thealgorithm. When making the choice, it is worthwhile to considerhow close is to and how close

is to .When constructing codes in this manner, the search for new

codewords in each step is conveniently handled via cliques incertain graphs. Assume that a code with constant weightis lengthened by adding a coordinate with a 1 to the codewordsand then including codewords with a 0 in the new coordinate(the alternative case is handled analogously). The compatibilitygraph consists of one vertex for each word with a 0 in the newcoordinate that has weight and is at distance at least from allcodewords of . There is an edge in the compatibility graph forevery pair of vertices whose corresponding words are at mutualdistance at least . Cliques in the compatibility graph now givedesired codes. The Cliquer software [34] was used in the currentwork to find cliques.

The approach of classifying constant weight codes via sub-codes has earlier been used by Brouwer et al. in [10], mainlyto prove nonexistence of codes and thereby to improve upperbounds on .

If the length is large and the size is small, then the out-lined approach might be infeasible due to large compatibilitygraphs or an abundance of cliques (because of large automor-phism groups of the codes). The larger the minimum distance

, the more accentuated is the need for an alternative approachfor handling such situations. The following method for classi-fying constant weight codes codeword by codeword is such anapproach.

In the codeword-by-codeword approach one simply addsone codeword at a time and carries out isomorph rejection

throughout the search. The only requirement on the new code-word is that its distance to previously included codewords mustbe at least .

The codeword-by-codeword approach is a slight modifica-tion of a standard method for classifying designs with ,also known as 2-designs or balanced incomplete block designs(BIBDs), in a point-by-point fashion [22, Section 6.1.1]. Thecodewords of a constant weight code can be viewed as vectorsand the minimum distance criterion as a requirement that theinner product between two vectors be at most . The sit-uation where the inequality for the inner products is replacedby equality is encountered when constructing BIBDs point bypoint.

Indeed, there is a one-to-one correspondence between BIBDsand optimal constant weight codes for parameters for which theformer objects exist [39], so classification results for BIBDs im-mediately give classification results for constant weight codes.It should be emphasized that this correspondence differs fromthat outlined in the Introduction for -designs in that there isnow one codeword in the constant weight code for each point(rather than block) of the BIBD. The correspondence can be ex-pressed as in the following theorem (see [39], [10, Theorem 9],and [22, Corollary 2.89] for details).

Theorem 2: There is an optimal constant weightcode of size that fulfills

iff there is a 2- design.Petrenjuk [37] used the codeword-by-codeword ap-

proach—in the framework of packings—for classifying theconstant weight codes of any size , thereby

also obtaining a classification of the constant weightcodes with and arbitrary size. It is erroneously claimedon [37, p. 245] that (the correct value is 6). Thecodes listed in [37] were not checked in the current work; thementioned error may be due to an erroneous list.

The codeword-by-codeword approach was also used byMayhew and Royle [28], to classify the constantweight codes of any size . A classification of suchcodes in the current work corroborate the tabulated numbers,except for the numbers of codes with 1, 12, 13, 15, 16, and17 codewords, for which the counts 1 (the original number2 is an apparent typo), 197898109, 416277783, 720126835,579539502, and 329728134, respectively, were obtained here.

Hybrid methods containing ingredients of both discussedmethods may sometimes be useful, for example, applying aclique algorithm to complete a code that has been constructedone codeword at a time.

III. ISOMORPH REJECTION

Isomorph rejection is a central part of any computationalwork on classifying combinatorial objects. Indeed, isomorphrejection is essential in intermediate steps to speed up thesearch and in the final stage to get an exhaustive collection ofobjects up to equivalence. Without a good approach and propertools, isomorph rejection can be a bottleneck in the endeavour,

ÖSTERGÅRD: BINARY CONSTANT WEIGHT CODES 3781

TABLE ISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 4

and improperly done it may introduce errors that cannot bediscovered easily without alternative computations. On theother hand, as we shall see, the search may be administrated sothat auxiliary data related to isomorph rejection can be used forvalidating the final results.

An exhaustive treatment of classification of combinatorial ob-jects can be found in [22]. An excellent tool for determiningequivalence of constant weight codes is the nauty software [29],which can be applied to colored graphs obtained from codes.

The construction of colored bipartite graphs for determiningequivalence of constant weight codes is folklore: There is a (say)blue vertex for each codeword (these vertices form the set )and a red vertex for each coordinate (these form ); the neigh-bors of a vertex (codeword) in are in and are given by thesupport of the codeword.

The straightforward method of carrying out isomorph rejec-tion by comparing a canonically labelled object produced bynauty with a list of previously encountered objects is limitedby the amount of memory available. The method of canonicalaugmentation, developed by McKay [30], however, is much lessrestricted by the amount of memory available (and in fact has aversion for which the amount of memory available is practicallyirrelevant). For further details about these and other isomorphrejection techniques, see [22, Ch. 4].

We shall now see how canonical augmentation can be uti-lized when classifying constant weight codes, focusing on thecase when constant weight codes are constructed from

constant weight codes. The case of insteadstarting from constant weight codes can be han-dled analogously.

From an constant weight code of size , onecan obtain constant weight subcodes byshortening; any of the coordinates can be deleted when formingsuch subcodes. Obviously, the constant weight codecould have been produced from any of these subcodes (someof which may be equivalent). We now want to particularizeone (orbit) of the subcodes, which will be called the canonicalparent of the code, and reject codes that are not constructedfrom the canonical parent; this is named the parent test.

Since there is a bijective correspondence between the sub-codes and the coordinates (deleted when forming the subcodes)of the final code, one may focus on the coordinates of the finalcode instead of the subcodes when determining the canonicalparent. Indeed, as nauty produces a canonical order of the ver-tices, this order may be used to pick out an orbit of vertices (con-sidering the vertices in ).

However, since the use of nauty is a comparatively expensiveway of carrying out the parent test, one should preferably in-clude one or several easily computed invariants in the definition

of the canonical parent, and use nauty as the last resort if morethan one candidate for the canonical parent remains.

Some basic invariants of constant weight codes are discussedin [37]. In the current work, it is among other things requiredthat the coordinate corresponding to the canonical parent havethe largest number of 1s. Consequently, if the search is startedfrom a subcode of size , then any solutions (final codes) thathave subcodes of size larger than can be immediately re-jected. It is crucial that this definition of the canonical parent isin accordance with the choice to carry out the search only fromcodes of size at least .

A code that passes the parent test might still be equivalent tocodes with the same canonical parent and therefore has to besubjected to a final test. In this test, one may use the automor-phism group of the subcode from which the search started andaccept a final code if it is, for example, lexicographically min-imum under the action of that group. In particular, this meansthat if the automorphism group is the trivial group, then the codecan be accepted immediately after the parent test is passed. Ifnauty was used in the parent test, then the canonical represen-tative produced by nauty can also be used for the final test. Theorder of the automorphism group of the accepted codes can beobtained simultaneously in either case.

Canonical augmentation is used here for isomorph rejectionalso in the codeword-by-codeword approach, where the addedcodeword is subjected to the parent test. Again, an invariant isutilized: Considering the sum of distances from a codeword tothe other codewords, the particularized (orbit of) codeword(s)should have the smallest value for this sum.

In the codeword-by-codeword approach, one may take thefollowing symmetry into account already when considering can-didates for codewords [11]: If there are consecutive coordi-nates where all codewords coincide, then if the new codewordhas 1s within these coordinates, up to symmetry one needconsider only one such subset of 1s. Note, however, that as thetotal number of codeword candidates will be required in theprocess of validation, discussed in Section IV, a codeword of the

aforementioned kind should be counted as candidates; we

get a product of such values if there are many sets of equal coor-dinates. (If isomorph rejection is carried out based on the actionof the automorphism group of the subcode, then the candidatethat survives that test must coincide with the chosen subset of1s.)

IV. VALIDATION OF RESULTS

Several recent classification results have been validated bya double counting argument using data from the classification,including [21], [36]; see [22, Ch. 10] for variants and further


reading. From a collection of representatives from all equiv-alence classes of constant weight codes with givenparameters and size —the subsequent techniques work foroptimal as well as suboptimal codes—one gets the number oflabelled such codes by the orbit-stabilizer theorem

(1)

On the other hand, by considering an arbitrary labelled codeand its subcode corresponding to the 1s in the first coordinate,one realizes that the total number of labelled codes can also beobtained as

(2)

where is a collection of representatives from all equivalenceclasses of codes and is the number offinal codes (before isomorph rejection) that are obtained (in thecomputer search) starting from .

We should now get the same value for (1) and (2), but we haveto modify this argument as only the codes ofsize at least are considered in our computer search.The number of labelled codes whose first coordinate has at least

1s is

(3)

where is the total number of coordinates of with at least1s. The sum (3) should be equal to (2) when is

a collection of representatives from all equivalence classes ofconstant weight codes of size at least .

Combining these results, we get the final test

(4)

all values of which can be easily collected during the search.The case when constant weight codes are con-

structed from constant weight codes is handledanalogously.

Validation based on double counting is also possible whenconstructing codes codeword by codeword. Denoting a collec-tion of representatives of the equivalence classes ofconstant weight codes of size by , consider adding onecodeword to the codes in to get the constant weightcodes of size . Moreover, let denote the total number ofdifferent words that can be added to a particular code(recall the comment in the last paragraph of Section III). Then

which simplifies to

(5)

TABLE IISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 6

TABLE IIISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 8

The GMP software [13] was used to carry out the calculationsof (4) and (5) precisely.

V. TABLES OF BOUNDS AND CLASSIFICATION RESULTS

Exhaustive classification results for short constant weightcodes are displayed in Tables I to VIII. For each even

, the values of (on the left, taken from[1]) and the number of optimal codes (on the right) are tabulatedfor , starting from the shortest length forwhich there are optimal codes of size greater than 3.

It is trivial to classify codes of size 1 and 2, and codes ofsize 3 can be classified easily using the approach in [37] (whichalso contains explicit classification results for size 3 over a widerange of parameters). There are unique optimal codes for

, and is handled by the one-to-one correspondencebetween the and the constant weightcodes.

One single entry in the tables is still open, the number of codesattaining (obtained in [1]) in Table IV. Allother subcases of this instance are handled easily, except for thecase with nine 1s in each coordinate (that is, all subcodes withrespect to the 1s have size 9).

References are given for all classification results in the right-hand side of Tables I to VIII. The unmarked values are obtainedin the current work; the origins of the other results—most ofwhich follow from classifications of BIBDs—are indicated bythe following letters:

Barrau [2] (Steiner system)

Best [3]; see also [5]

Best [4]; see also [5]

Petrenjuk [37]

Basic BIBD; see [22, Section 6.1.6]

Martinetti [26], Todd [43] (BIBD)

Nandi [33] (BIBD)

Husain [17] (BIBD)


TABLE IVSIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 10

TABLE VSIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 12

TABLE VISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 14

TABLE VIISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 16

TABLE VIIISIZE AND NUMBER OF OPTIMAL CODES WITH MINIMUM DISTANCE 18

Gibbons, Mathon, and Corneil [12], Stanton, Mullin,and Bate [41], Van Lint, Van Tilborg, and Wiekema[24], Breach [8] (BIBD)

Hall, Jr. [16], Bhat [7] (BIBD)

Kaski and Östergård [20] (BIBD)

Morgan [32] (BIBD)

Gronau and Prestin [15] (BIBD)

Breach [27] (BIBD)

Pietsch [27] (BIBD)

Ito, Leon, and Longyear [18]; and Kimura [23] (BIBD)

Gibbons, Mathon, and Corneil [12], Van Lint, VanTilborg, and Wiekema [24] (BIBD)

Brouwer [9]

Todd [43] (BIBD)

For the smallest weights, it is possible to extend the classifica-tion outside the ranges of Tables I to VIII. Several classificationresults for BIBDs leading to results for longer codes are alsoknown; see [22, Sec. 6.1.6] and [27].

The current work has not only led to classification results butalso to new upper bounds on , which have even im-plied some exact values of . No new lower bounds(improved codes) were encountered in the study.

One new exact value of is indicated in the left-hand side of Table I, and updated best known upper bounds fora range of parameters are given in Tables IX, X, and XI for

, and , respectively. The exact values in Tables IX to XIare indicated by a period. Original references for the unmarkedbounds can be found in [1]; the origins of the other bounds areshown by the following capital letters:

This work

Johnson bound (Theorem 1)


TABLE IXBEST KNOWN UPPER BOUNDS ON ��

TABLE XBEST KNOWN UPPER BOUNDS ON ��

TABLE XIBEST KNOWN UPPER BOUNDS ON ��

Östergård and Pottonen [35] (Nonexistence of Steinersystem), [1, Theorem 13]

Schrijver [38]

The determined exact values are as follows.

Theorem 3:,

and .

Proof: The only listed result that is not obtained by di-rectly applying a method considered in Sections II and III is

.With the parameters , and, one gets and ; since

and , the choice is made tocarry out the classification via the former alternative. There are11 inequivalent codes attaining —this is an oldresult [3]—and a recursive classification via subcodes revealsthat there are 1225 inequivalent codes with the same parametersand 34 codewords. Unfortunately, the clique search in the finalstep of the classification—which turns out to be a nonexistenceproof—is prohibitively slow, and other techniques are requiredto search for sets of and codewords.

Consider the optimal codes attaining andfocus on one particular such code . The remaining

codewords to be found have a 0 in the new coordinate, andthey must have a coordinate with at least

1s. As an intermediate step in the search, for each coordinateand each possible size we search for all possible

ways of lengthening by adding a coordinate with a 1 for thewords of and including codewords that have a 0 in the newcoordinate and a 1 in coordinate . Isomorph rejection may becarried out subsequently, partially or totally. This intermediate(clique) search as well as the very final (clique) search for theremaining codewords are reasonablyfast.

The codes attaining lead to intermediatesolutions with . The cases with a subcode of size34 can be handled in an analogous way. The search reveals that

and, since it is known [10] that, that .

The technique in the proof of Theorem 3 was further usedto classify the optimal codes attaining andto prove the bounds and .Since it is known [1] that , we now have the firstknown set of parameters for whichwith ; this settles [25, Research Problem A5].

Complete catalogues of all BIBDs to which references aregiven here are available in an electronic supplement to [22]. Theauthor is happy to provide lists of any of the code families clas-sified here in electronic form (the work has also resulted in clas-sification results for many types of suboptimal codes).

About two months of CPU time of a one-core 2-GHz per-sonal computer was needed to obtain the current classification(most of the time was used for just a couple of entries). A moreextensive use of computer time would make it possible to (justslightly) push the limit further with the current approach.

ACKNOWLEDGMENT

The author would like to thank A. Brouwer and the anony-mous referees for comments and corrections.

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Patric R. J. Östergård was born in Vaasa, Finland, in 1965. He received theM.Sc. (Tech.) degree in electrical engineering and the D.Sc. (Tech.) degree incomputer science and engineering, in 1990 and 1993, respectively, both fromHelsinki University of Technology TKK, Espoo, Finland.

From 1989 to 2001, he was with the Department of Computer Scienceand Engineering, TKK. During 1995–1996, he visited Eindhoven Universityof Technology, The Netherlands. Since 2000, he has been a Professor withTKK—which merged with two other universities to form the Aalto Universityin January 2010—with the Department of Communications and Networking.He was the Head of the Communications Laboratory, TKK, during 2006–2007.He spent six months with the Universität Bayreuth, Germany, in 2010. Heis the coauthor of Classification Algorithms for Codes and Designs (Berlin:Springer-Verlag, 2006). His research interests include algorithms, codingtheory, combinatorics, design theory, and optimization.

Dr. Östergård is a Fellow of the Institute of Combinatorics and its Applica-tions. He is a recipient of the 1996 Kirkman Medal. Since 2006, he has been aco-Editor-in-Chief of the Journal of Combinatorial Designs.

classification of binary constant weight codes

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