improved semidefinite programming bound on sizes of codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 11, NOVEMBER 2013 7337

Improved Semidefinite Programming Boundon Sizes of CodesHyun Kwang Kim and Phan Thanh Toan

Abstract—Let (respectively ) be themaximumpossible number of codewords in a binary code (respectively, bi-nary constant-weight code) of length and minimum Hammingdistance at least . By adding new linear constraints to Schrijver’ssemidefinite programming bound, which is obtained from block-diagonalizing the Terwilliger algebra of the Hamming cube, we ob-tain two new upper bounds on , namely and

. Twenty three new upper bounds on forare also obtained by a similar way.

Index Terms—Binary codes, binary constant-weight codes,linear programming, semidefinite programming, upper bound.

I. INTRODUCTION

L ET and let be a positive integer. The (Ham-ming) distance between two vectors in is the number

of coordinates where they differ. The (Hamming) weight of avector in is the distance between it and the zero vector. Theminimum distance of a subset of is the smallest distance be-tween any two different vectors in that subset. An code isa subset of having . If is ancode, then an element of is called a codeword and the numberof codewords in is called the size of .The largest possible size of an code is denoted by

. The problem of determining the exact values ofis one of the most fundamental problems in com-

binatorial coding theory. Among upper bounds on ,Delsarte’s linear programming bound is quite powerful (see [1]and [2]). This bound is obtained from block-diagonalizing theBose–Mesner algebra of . In 2005, by block-diagonalizingthe Terwilliger algebra (which contains the Bose–Mesneralgebra) of , Schrijver gave a semidefinite programmingbound [3]. This bound was shown to be stronger than or as goodas Delsarte’s linear programming bound. In fact, 11 new upperbounds on were obtained in the paper for . In2002, Mounits et al. added more linear constraints to Delsarte’slinear programming bound and obtained new upper bounds on

Manuscript receivedDecember 14, 2012; revised July 04, 2013; accepted July28, 2013. Date of publication August 15, 2013; date of current version October16, 2013. H. K. Kim was supported by Basic Research Program through theNational Research Foundation of Korea (NRF) funded by the Ministry of Edu-cation, Science, and Technology under Grants 2010-0026473 and 2012047640.P. T. Toan was supported by Basic Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education, Sci-ence, and Technology under Grant 2010-0026473.The authors are with the Department of Mathematics, Pohang Uni-

versity of Science and Technology, Pohang 790-784, Korea (e-mail:[email protected]; [email protected]).Communicated by N. Kashyap, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2013.2277714

[4]. In this paper, we construct new linear constraintsand show that these linear constraints improve Schrijver’ssemidefinite programming bound. Among improved upperbounds on for , there are two new upperbounds, namely and .An constant-weight code is an code such

that every codeword has weight . Let be thelargest possible size of an constant-weight code. Theproblem of determining the exact values of hasits own interest. Upper bounds on can even helpto improve upper bounds on (for example, see [2],[4]). There are also Delsarte’s linear programming bound andSchrijver’s semidefinite programming bound on[1], [3]. In 2000, Agrell et al. added new linear constraints toDelsarte’s linear programming bound and improved severalupper bounds on [5]. More linear constraints thatimprove upper bounds on can be found in [6]. Inthis paper, we add further new linear constraints to Schrijver’ssemidefinite programming bound on and obtain 23new upper bounds on for .

II. UPPER BOUNDS ON

In this section, we improve upper bounds on byadding more linear constraints to Schrijver’s semidefinite pro-gramming bound, which is obtained from block-diagonalizingthe Terwilliger algebra of the Hamming cube . For moredetails about Schrijver’s semidefinite programming bound, see[3].

A. General Definition of and

We first give a general definition. Let and be positive inte-gers. For a finite (possibly empty) set , whereeach is a vector in and each is a nonnegative integer,we let be the maximum possible number of code-words in a binary code of length andsuch that each codeword is at distance from for all .1) : If is empty, then we get the usual definition of

.2) : If contains only one element, says ,

then is the maximum possible number of codewordsin a binary code of length and minimum distance suchthat each codeword is at distance from . By translation, wemay assume that is the zero vector so that each codeword hasweight . Therefore, , where .A doubly-constant-weight code is an

constant-weight code such that everycodeword has exactly ones on the first coordinates(and hence has exactly ones on the last coordinates).Let be the largest possible size of a

0018-9448 © 2013 IEEE

7338 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 11, NOVEMBER 2013

doubly constant-weight code. Agrell etal. showed in [5] that upper bounds oncan help improving upper bounds on . In our re-sult, upper bounds on will be used toimprove upper bounds on . As and ,

is also a special case of .3) : If contains two elements, then the

following proposition shows that is exactly.

Proposition 1: If , then

(1)

where ,and .

Proof: Let and . Bytranslation, we may assume that is the zero vector. Hence,

. Let be a vector at distance fromand at distance from . By rearranging the coordinates,

we may assume that

Since is the zero vector, we have

(2)

Also,

(3)

Equations (2) and (3) give and.

4) : It becomes more complicated when con-tains more than two elements. We consider a very specialcase when , which will be used in our improvingupper bounds on in Section III. Suppose that

satisfies thefollowing conditions.1) is the zero vector (which can always be assumed).2) and have the same weight .3) .Then, ,where and are determined in the next propo-sition. The definition of issimilar to that of (it is the largest possiblesize of a code such that on each codeword thereare exactly ones on the coordinates ).Proposition 2: Suppose that satisfies

is the zero vector, , and .Then,

(4)

where , ,, ,

, ,and .

Proof: Suppose that is a vector at distance from. By rearranging the coordinates, we may

assume the following.

Let be as in the above figure. Sinceand have the same weight,. Now . Therefore,

and . We have

.

Solving these equations, we obtain as desired.

B. Schrijver’s Semidefinite Programming Bound on

Let be the collection of all subsets of . Eachvector in can be identified with its support (the support of avector is the set of coordinates at which the vector has nonzeroentries). With this identification, a code is a subset of andthe (Hamming) distance between two subsets and in is

. Let be an code. For each , and, define

(5)

where denotes the number of pairwise disjointsubsets of sizes , respectively, of a set of size ,and denotes the number of triples with

, and ,or equivalently, with , , and

. Set if .The key part of Schrijver’s semidefinite programming bound

is that for each , the matrices

(6)

and

(7)

are positive semidefinite, where is given by

(8)

KIM AND TOAN: IMPROVED SEMIDEFINITE PROGRAMMING BOUND ON SIZES OF CODES 7339

Since , an upper bound on can beobtained by considering the as variables and by

(9)

subject to the matrices (6) and (7) are positive semidefinite foreach and subject to the following conditionson the (see [3]).i) .ii) and for all

.iii) if is a permutation of

.iv) if .

C. Improved Schrijver’s Semidefinite Programming Boundon

1) New Constraints for : Let be an code and letbe defined by (5).

Theorem 3: For all with,

(10)

Proof: Recall that is the number of tripleswith , , and .

For any pair with , the number ofsuch that and is upper

bounded by , where .By Proposition 1, . Sincethe number of pairs such that is ,

. Therefore,

(11)

The following corollary was used in [3].Corollary 4: For each ,

(12)

Proof: By Theorem 3, .

Remark 5: Theorem 3 improves the conditionin Schrijver’s semidefinite programming bound since

(in fact, is much less

than 1 in general). Similarly, Corollary 4 in many cases (ofand ) improves the condition since

is much less than in general.

2) Delsarte’s Linear Programming Bound and Its Improve-ments: Let be an code, the distance distribution

of is defined by

(13)

By definition, for each . Hence,is the distance distribution on . The following

result can be found for example in [7] or [6].Theorem 6: (Delsarte’s linear programming bound and its

improvements). Let be an code with distance distribu-tion . For ,

(14)

where is the Krawtchouk polynomial given by

(15)

If is odd, then

(16)

If (mod 4), then there existssuch that

(17)

3) Linear Constraints on Distance Distributions :If some linear constraints are used to improve Delsarte’s linearprogramming bound on , then these constraints can stillbe added to Schrijver’s semidefinite programming bound to im-prove upper bounds on . The following constraints aredue to Mounits et al. (see [4, Ths. 9 and 10]).Theorem 7: Let be an code with distance distribution

. Suppose that is even and . Then,

(18)

and

(19)

for all .Table I shows improved upper bounds on when

linear constraints in Theorems 3, 6, and 7 are added to Schri-jver’s semidefinite programming bound (9). In the table,by Schrijver bound we mean upper bound obtained fromSchrijver’s semidefinite programming bound (9). Amongimproved upper bounds on , there are two newupper bounds: and . Asin [3], all computations here were done by the algorithmSDPT3 available online on the NEOS Server for Optimization(http://www.neos-server.org/neos/solvers/index.html). Since

if is odd, we can always assume


TABLE IIMPROVED OPPER BOUNDS FOR

that is even. If is even, then is attained by a codewith all codewords having even weights. Hence, when per-forming computations of Schrijver’s semidefinite programmingbound, one can put if or is odd.Remark 8: Delsarte’s linear programming bound is from

linear constraints on the distance distribution on pairs of code-words. Schrijver obtained semidefinite programming bound(9) by exploiting constraints on triples of codewords [3]. In[8], Gijswijt et al. extended the semidefinite programmingbound by exploiting constraints on -tuples of codewords andworked out the corresponding bounds for quadruples ( ).In Table I, all best upper bounds previously known (includingthe bound ) follow from their work. Note that ittook even months to finish computing of an upper bound usingthe quadruple-distance semidefinite programming bound whiletogether with our improvements, the semidefinite programmingbound (9) requires only a small amount of time of performingcomputations (at most some minutes for each upper bound).Remark 9: In Theorems 3 and 7, the values of

and may have not yet been known. How-ever, we can replace them by any of their upper bounds (see theproof of [4, Th. 10] for the validity of this replacement in The-orem 7). While best-known upper bounds on (whichare mostly from [3], [5], [9], [10]) are used in our computa-tions, all upper bounds on that we used arefrom the tables on Erik Agrell’s website http://webfiles.portal.chalmers.se/s2/research/kit/bounds/dcw.html.

III. UPPER BOUNDS ON

A. Some Properties of

We begin with some elementary properties ofwhich can be found in [2].Theorem 10:

(20)

(21)

(22)

(23)

(24)

Remark 11: By (20) and (22), we can always assume thatis even and . Also, by (21), (23), and (24), we can assumethat .

B. Schrijver’s Semidefinite Programming Bound on

Let be an constant-weight code and let .For each , , , and , define

(25)

where is the number of triples with, and

, or equivalently, with , ,, and . Set

if either or .In the previous section, depends on . Hence,

should be denoted by . We will use the later notation inthis section. As in [3], for each and each

, the matrices

(26)

and

(27)

are positive semidefinite, where and. Since ,

an upper bound on can be obtained by consideringthe as variables and by

(28)

subject to the matrices (26) and (27) are positive semidefinitefor each and each , andsubject to the following conditions.1) .2) and for all

.3) if andis a permutation of .

4) if .

C. Improved Schrijver’s Semidefinite Programming Boundon

1) New Constraints for : Let be an constant-weight code and let be defined by (25). The following the-orem corresponds to Theorem 3 in the previous section.Theorem 12: For all with

and ,

(29)

Proof: Suppose that such that .We claim that the number of codewords suchthat , , and


is upper bounded by. It is easy to see that this number is upper

bounded by , where. By Proposition 2,

, where, ,

, and similarly,.

Hence,, where the

later equality comes from Proposition 23 (iii) in the Appendix.Since the number of pairs such thatis , .Therefore,

2) Delsarte’s Linear Programming Bound: Let be anconstant-weight code with distance distribution

. By definition of , for every(note that and whenever is odd oror ).Theorem 13 (Delsarte’s linear Programming Bound): If

is the distance distribution of an con-stant-weight code, then for ,

(30)

where

(31)

Specifying Delsarte’s linear programming bound ongives the following linear constraints on , which sometimeshelp reducing upper bounds on by 1 (see [6, Propo-sition 11]).Theorem 14: Let be an constant-weight code with

distance distribution . For each ,

(32)

where and are the quotient and the remainder, respectively,when dividing by , i.e.,

(33)

with , and where is defined by

(34)

3) New Linear Constraints on Distance Distributions: Linear constraints which correspond to those in The-

orem 7 have not been studied for constant-weight codes eventhough similar constraints have been studied by Argrell et al.[5] (see Theorem 21 below). We now present these constraints.Several new notations are needed. For convenience, we fix thefollowing settings until the end of this section.1) is an constant-weight code with distance distri-bution such that is even and .

2) Let . Since , .3) Let , which is the set of allpositive integer such that can be nonzero.

4) For each , let be the set of all vectors insuch that has exactly ones on the first coordinatesand exactly ones on the last coordinates.

5) For both in , define.

6) For each codeword in , let, which is the set of all codewords in

at distance from . By definition of ,for each .

7) For each , let denote an integer such that.

8) For both in with and , letdenote an integer such that .

Proposition 15: For both in , , where

ifif

and

ifif

In particular, if , then .Proof: The proof is straightforward.

Lemma 16: For each and each codeword ,

(35)

Proof: Let be a codeword in . It is easy to seethat is upper bounded by , where

. By Propositions 1 and 23(iii),. Hence,

.Theorem 17: Suppose that is a nonempty subset of

such that for all both in . Then, for eachcodeword , is nonempty for at most one in .Furthermore,

(36)


Proof: Let be a codeword in . Suppose on the contrarythat there exist both in such that andare nonempty. Then, choose any and .By rearranging the coordinates, we may assume that

(37)

Since and and have the same weight ,must have exactly ones on the first coordinates and

exactly ones on the last coordinates. This means .Similarly, . By definition of ,

. Thus, ,which is a contradiction since and are two different code-words in . Hence, is nonempty for at most one in .It follows by Lemma 16 that

(38)

Taking sum of (38) over all , we get .We now consider the case for some both in .

The following Lemma says that the existence of a codeword atdistance from may reduce the total number of codewordsat distance from .Lemma 18: Suppose both in such that and

. If is a codeword in such that , then.

Proof: Fix a codeword . If is empty,then there is nothing to prove. Hence, we assume. Let . By rearranging the coordinates, we mayassume that

(39)

As in the proof of Theorem 17, we can show thatand . By definition of ,

. Thus,.

Since , by rearranging the first coordinates,we may assume that on the first coordinates:

(40)

On the first coordinates, must have exactlyones on the first coordinates (the other ones ofmust be fixed since ).Similarly, since , by rearranging the last coordi-

nates, we may assume that on the last coordinates

(41)

On the last coordinates, must have exactlyones on the first coordinates (the other ones ofmust be fixed since ).

From (39), (40), and (41), we obtain

(42)

Now the number of is upper bounded by, where

. By Proposition 15, . ApplyingProposition 2 (and using and ),we obtain

where the last equality comes from Proposition 23 inthe Appendix. Therefore,

.Theorem 19: Suppose that is a subset of satisfying the

following properties.1) .2) There exist both in such that and

.3) For all both in such that either or

, we always have .Let . Then,

(43)

(44)

(45)

Proof: We first prove (43). It suffices to show that for everycodeword in ,

(46)

if . Let be any codeword in . By Lemma16, and . Also by Lemma 18,

if and if. We prove (46) by considering the following three

cases.Case 1: : Proving (46) is exactly the same as

proving (38). So we are done.Case 2: and : Since

, for every by The-orem 17. Hence, to prove (46), we only need to provethat . By hypothesis,


. Thus, and hence.

Case 3: and : As in Case 2,for every . We have

.Therefore, (46) is proved and so is (43).By symmetry, (44) follows.We now prove (45). It suffices to show that for every code-

word in ,

(47)

if . If either or ,then proving (47) is exactly the same as proving (38). Hence,suppose that and . As in Case 2,

for every . We have

(48)

We now specify which are used in Theorems 17 and 19.Let , so that

. Also, let .

1) Case 1: is even. In this case, . We apply Theorem17 for and apply Theorem 19 for

(withand ) for each .

2) Case 2: is odd. In this case, . We apply Theorem19 for (with

and ) for each .Example 20: Consider . We have

is odd. Hence, and . So,and . We can apply Theorem 19 for

(with ). We have

and

for . Since , Theorem 19

gives and . Thelater constraint is equivalent to .In fact, adding these two linear constraints to Delsarte’s linearprogramming bound (Theorem 13), we get the new upper bound

This improves the upper bound in [5] andthe best (previously) known upper bound

in [3]. For (with ), Theorem 19 gives lesseffective linear constraints.When , there is no set satisfying Theorem

19. In this case, the following type of linear constraintswhich comes from [5, Proposition 17] is very useful. Asin [5], let be the largest possiblesize of a doubly-bounded-weight code(a doubly-bounded-weight code is an

constant-weight code such that everycodeword has at most ones on the first coordinates). Ta-bles for upper bounds on can be found onErik Agrell’s website http://webfiles.portal.chalmers.se/s2/re-search/kit/bounds/dbw.html.Theorem 21: Let . For with. If , define and as any nonnegative

integers such that

(49)

(50)

where . Also, define for each .Then,

(51)

(52)

(53)

By adding the linear constraints in Theorems 12, 14, 17, 19,and 21 to Schrijver’s semidefinite programming bound (28), weobtained new upper bounds on shown in Table II.As before, all computations were done by the same algorithmSDPT3 at the same server.Remark 22: The new upper bounds on presented

in Table II do not give rise to any further improved upper boundson in the range through either the improvedsemidefinite programming bound in Section II or known in-equalities relating the two quantities such as the well-knownElias bound (for example, see [2, pp.

558]), the Johnson bound (see [11, Th. 1]), and even their im-provements in [12] and [4, Th. 4], respectively.

APPENDIXUPPER BOUNDS ON

To apply Theorem 12, we need tables of upper bounds on. However, there are no

such tables available since this is the first time the functionis introduced. We show

here some elementary properties that are used to obtain upperbounds on .


TABLE IINEW UPPER BOUNDS FOR

In general, let us define as follows.For , a multiply-constant-weight codeis a code such that there are exactly oneson the coordinates. When this is definition of an

constant-weight code, when this is defi-nition of a doubly-constant-weight code,etc. Let be the largest possible size of a

multiply-constant-weight code.We present here elementary properties that are used

to get upper bounds on . The proofsof these properties are similar to those for or

, and hence are omitted. Upper bounds onthat we used in Theorem

12 are the best upper bounds obtained from these properties.Proposition 23:i) If is odd then,

(54)

ii) If for some , then

(55)

iii) does not change if we replace anyby .

iv) .

v) .

vi) if .

Remark 24: By (i) and (iv), we can always assume thatis even and . By (ii) and (iii), we may assume that

for each . Also, by (v) and (vi), we can assumethat .The next proposition can be used to reduce the size of

from to . When the size of the set is 2, weuse known upper bounds on .


(56)

where for , and.


(57)

where is obtained from by re-placing the pair by .


(58)

where is obtained from by re-placing the pair by .

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their comments and suggestions, which not only improvedthe presentation of the results but also made the paper moreinformative.

REFERENCES[1] P. Delsarte, “An algebraic approach to the association schemes of

coding theory,” Philips Res. Rep. Suppl., no. 10, 1973.[2] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting

Codes. Amsterdam, The Netherlands: North-Holland, 1977.[3] A. Schrijver, “New code upper bounds from the Terwilliger algebra

and semidefinite programming,” IEEE Trans. Inf. Theory, vol. 51, no.8, pp. 2859–2866, Aug. 2005.

[4] B. Mounits, T. Etzion, and S. Litsyn, “Improved upper bounds on sizesof codes,” IEEE Trans. Inf. Theory, vol. 48, no. 4, pp. 880–886, Apr.2002.

[5] E. Agrell, A. Vardy, and K. Zeger, “Upper bounds for constant-weight-codes,” IEEE Trans. Inf. Theory, vol. 46, no. 7, pp. 2373–2395, Nov.2000.

[6] B. G. Kang, H. K. Kim, and P. T. Toan, “Delsarte’s linear programmingbound for constant-weight codes,” IEEE Trans. Inf. Theory, vol. 58, no.9, pp. 5956–5962, Sep. 2012.

[7] M. R. Best, A. E. Brouwer, F. J. MacWilliams, A. M. Odlyzko, and N.J. A. Sloane, “Bounds for binary codes of length less than 25,” IEEETrans. Inf. Theory, vol. IT-24, no. 1, pp. 81–93, Jan. 1978.

[8] D. C. Gijswijt, H. D. Mittelmann, and A. Schrijver, “Semidefinite codebounds based on quadruple distances,” IEEE Trans. Inf. Theory, vol.58, no. 5, pp. 2697–2705, May 2012.

[9] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, andW. D. Smith, “A newtable of constant weight codes,” IEEE Trans. Inf. Theory, vol. 36, no.6, pp. 1334–1380, Nov. 1990.

[10] P. R. J. Östergård, “Classification of binary constant weight codes,”IEEE Trans. Inf. Theory, vol. 56, no. 8, pp. 3779–3785, Aug. 2010.

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[12] N. Q. A. L. Györfi and J. L. Massey, “Constructions of binary constant-weight cyclic codes and cyclically permutable codes,” IEEE Trans. Inf.Theory, vol. 38, no. 3, pp. 940–949, May 1992.


Hyun Kwang Kim received the B.S. and M.S. degrees in mathematics fromSeoul National University, Seoul, Korea, in 1979 and 1981, respectively, andthe Ph.D. in mathematics from the Johns Hopkins University, Baltimore, MD.Since 1988, he has been with the Department of Mathematics, Pohang Uni-

versity of Science and Technology, Pohang, Korea. From 1999 to 2000, hevisited the School of Mathematics, Korea Institute of Advanced Study, Seoul,Korea. His research interests are subject related to number theory, coding theory,and combinatorics.

Phan Thanh Toan received the B.S. degree in mathematics and computersciences in 2004 from Vietnam National University-Ho Chi Minh City, HoChi Minh City, Vietnam, and the Ph.D. degree in mathematics in 2012 fromPOSTECH, Pohang, Republic of Korea.Since 2012, he has been a Postdoctoral Researcher at Department of Math-

ematics, POSTECH, Pohang, Republic of Korea. His research interests arecoding theory and commutative ring theory.

improved semidefinite programming bound on sizes of codes

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