2706 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, …staff.ustc.edu.cn/~drzhangx/papers/ternary cwc.pdf · In Section II, basic notations and results in coding theory and combinatorial

2706 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 5, MAY 2012

Optimal Ternary Constant-Weight CodesWith Weight 4 and Distance 5

Hui Zhang, Xiande Zhang, and Gennian Ge

Abstract—Constant-weight codes (CWCs) play an importantrole in coding theory. The problem of determining the sizes foroptimal ternary CWCs with length , weight 4, and minimumHamming distance 5 (� � �� code) has been settled for allpositive integers �� or �� and � �� with

� �� undetermined. In this paper, we investigate theproblem of constructing optimal � � �� codes for all lengths

with the tool of group divisible codes. We determine the size ofan optimal � � �� code for each integer � leaving thelengths �� unsolved.

Index Terms—Constant-weight codes (CWCs), group divisiblecodes (GDCs), ternary codes.

I. INTRODUCTION

C ONSTANT-WEIGHT codes (CWCs) play an importantrole in coding theory [26]. Binary CWCs have been ex-

tensively studied by many authors with the focus of attentionon the function , which denotes the maximum cardi-nality of a binary code of length , minimum Hamming distance

, and constant weight .Nonbinary CWCs have not received the same amount of

attention, but there have been a number of papers dealingwith this topic, see for example [3]–[5], [7], [11], [12], [25],[29], [32]–[36], and [40], due to several important applicationsrequiring nonbinary alphabets, such as the coding for band-width-efficient channels [9] and the design of oligonucleotidesequences for DNA computing [24], [27].

In [29], some methods for providing upper and lower boundson the maximum cardinality of a ternary code withlength , minimum Hamming distance , and constant weight

were presented, and a table of exact values or bounds inthe range was also given. We list the exact values of

for codes with length no greater than 10 in Table I.

Manuscript received December 02, 2010; revised June 17, 2011; acceptedNovember 22, 2011. Date of publication January 5, 2012; date of current versionApril 17, 2012. The work of H. Zhang was supported by the Scholarship Awardfor Excellent Doctoral Student granted by the China Ministry of Education. Thework of G. Ge was supported in part by the National Outstanding Youth ScienceFoundation of China under Grant 10825103, in part by the National NaturalScience Foundation of China under Grant 61171198, the Specialized ResearchFund for the Doctoral Program of Higher Education, and in part by the ZhejiangProvincial Natural Science Foundation of China under Grant D7080064.

The authors are with the Department of Mathematics, Zhejiang Univer-sity, Hangzhou 310027, Zhejiang, China (e-mail: [email protected];[email protected]; [email protected]).

Communicated by M. Blaum, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2011.2179412

TABLE IBOUNDS ON � �� FOR � � ��

Generalized Steiner systems were first intro-duced by Etzion [10] and used to construct optimal CWCs overan alphabet of size with minimum Hamming distance

, in which each codeword has length and weight . Alot of work has been done on the existence of a ;see, for example, [2], [8], [10], [13]–[15], [17], [21]–[23], [30],[31], [37], and [39]. In [23], Ji et al. proved that aexists if and only if and with the pos-sible exceptions . Equivalently, we have thefollowing lemma.

Lemma 1.1: For any integer , , and, there exists an optimal code with

codewords.The concept of group divisible codes (GDCs), an analog of

group divisible designs (GDDs) in combinatorial design theory[16], was first introduced by Chee et al. in [6]. This new class ofcodes is shown to be useful in recursive constructions of CWCsand constant-composition codes. In [40], GDCs have played asignificant role in constructing CWCs with weight 4 and min-imum Hamming distance 6. In this paper, we continue to inves-tigate the construction of optimal codes for all lengths

with the tool of GDCs. We determine the size of an optimalcode for each integer leaving eight lengths of

unsolved.This paper is organized as follows. In Section II, basic

notations and results in coding theory and combinatorial designtheory are given. In Section III, we shall construct some smallGDCs which will be used in the following sections. FromSections IV to VII, we will give the constructions for optimal

codes case by case. A brief conclusion will be givenin the last section.

II. PRELIMINARIES

For integers , the set is denoted by. The ring of integers modulo , , is denoted by ,

and the set of nonnegative integers is denoted by . For anytwo sets and , denotes the Cartesian product, i.e.,

.

A. -Ary CWCs

If and are finite sets, denotes the set of vectors oflength , where each component of a vector has value

0018-9448/$31.00 © 2012 IEEE

ZHANG et al.: OPTIMAL TERNARY CONSTANT-WEIGHT CODES WITH WEIGHT 4 AND DISTANCE 5 2707

in and is indexed by an element of , that is,and for each .

A -ary code of length is a set for some ofsize . The elements of are called codewords. The Hammingnorm or the Hamming weight of a vector is definedas . The distance induced by thisnorm is called the Hamming distance, denoted by , so that

, for . For any vector ,define the support of as .

A code is said to have (minimum Hamming) distance iffor all distinct . To simplify the presen-

tation, when we talk about the distance of a code, we alwaysmean the minimum Hamming distance of the code. Iffor every codeword , then is said to be of (constant)weight . A -ary CWC of length , distance , and weightis denoted as an code. The number of codewords inan code is called the size of the code. The maximumsize of an code is denoted as , and the

codes achieving this size are said to be optimal.The following bound for has been established by

Svanström [34].

Lemma 2.1 ([34]):

In the rest of this paper, we will use to denote

. It is well known that when; hence, we get the following upper bound.

Lemma 2.2: .

B. Designs

A set system is a pair such that is a finite set ofpoints and is a set of subsets of , called blocks. The orderof the set system is , the number of points.

For a set of nonnegative integers , a pairwise balanced de-sign ( -PBD) is a set system of order , such that

for all , and every pair of distinct elements ofoccurs in exactly one block of . If an element is

“starred” (written ), it means that the PBD has exactly oneblock of size .

Lemma 2.3 ([28]): There exists a -PBD withif and only if , and

1) or and or ; or2) or and or .Let be a set system and be a par-

tition of into subsets, called groups. The triple is aGDD when each pair of elements of not contained in a groupappears in exactly one block, and for all and

. Denote a GDD by -GDD if for all

. The type of the GDD is the multiset . An“exponential” notation is usually used to describe the type: type

denotes occurrences of , .A -GDD of type is also called a transversal design and

denoted by .

Lemma 2.4 ([1]): Let be a positive integer. Then1) a exists if ;

2) a exists if ;3) a exists if ;4) a exists if is a prime power.As stated in [10] and [39], a -ary CWC over

can be constructed from a given -GDD of type ,, where

and is the distance of the resulting code. For each block, we form a codeword of

length by putting in position , , and zeroselsewhere. A -GDD of type which forms a code withdistance is called a generalized Steiner system, denotedby .

The following definition of frame generalized Steiner systemwas first introduced by Ji et al. in [23] to construct generalizedSteiner systems.

Let and be a partition of intosubsets of size , . Suppose

is a -GDD of type .For each block , we ob-tain a codeword of length by putting in position ,

, and zeros elsewhere. If the resulting code has distance, we call such a -GDD a frame generalized Steiner

system (or shortly a frame) of type , and denote it byframe- . It is clear that ais also a frame- .

C. Group Divisible Codes

Given and , the constriction of to , written, is the vector such that .

Conversely, given and , the extension of to, written , is the vector such that

if ;if .

Given a set , let .A -ary GDC of distance is a triple , where

is a partition of with , and isa -ary code of length , such that for all distinct

, and for all , . Elements ofare called groups. We denote a GDC of distance

as - if is of constant weight . The type of a GDCis the multiset . As in the case of GDDs,

the exponential notation is used to describe the type of a GDC.The size of a GDC is . Note that an codeof size is also a -ary - of type and size .

The aforementioned notations were first introduced by Cheeet al. in [6]. By the definition of a frame-GS, we have the fol-lowing lemma.

Lemma 2.5: If there is a frame-with blocks, then there is a -ary - oftype and size .

Taking the frame-GSs given in [23], we have the followinglemma.

Lemma 2.6: There exist ternary 4- s of type andsize 60, type and size 120, type and size foreach , and type and size for each

.


The direct constructions of a code or a GDC in this paper arealways based on the familiar difference method, where a finitegroup (mostly abelian group ) will be utilized to generate allthe codewords of a code or a GDC. Thus, instead of listing all thecodewords, we list a set of base codewords and generate othersby an additive group or perhaps some further automorphisms.

To save space, for each codeword of a code ora GDC, we always list the set and use a subscript to de-note the value of for each . The automorphismgroup employed always acts on the supports of the base code-words with the subscripts fixed, if it is not mentioned.

Example 2.1: Let , and. Then, is a ternary 4- of type and size

120, where is the set of all cyclic shifts of the codewords:

Or equivalently, we can say that is obtained by developingthe elements of in the supports of following codewords

with the subscripts fixed:

It is easy to see that this ternary 4- of type isalso an optimal code since its size reaches the upperbound in Lemma 2.2.

Example 2.2: Let , and. Then, is a ternary 4- of type

and size 64, where is obtained by developing the elementsof in the following supports of codewordswith the subscripts fixed:

Example 2.3: Let , and. Then, is

a ternary 4- of type and size 276, whereis generated from the following codewords, whose

supports are developed under the automorphism groupwith

the subscripts fixed:

The following constructions are simple but useful, which aregiven in [6].

Construction 2.1 ([6]): (Fundamental Construction) Let, be a GDD, and be a

weight function. For any subset , let. Suppose that for each , there exists a -ary

- of typeand size . Then, is a -ary

- of type and size .

Example 2.4: Take a from Lemma 2.4and apply the Fundamental Construction with weight 4 to everypoint (that is, for any ). For each , thereis a ternary 4- of type and size 64 by Example 2.2(we call it an input GDC). The result is a ternary 4- oftype and size 1600.

Construction 2.2 ([6]): (Filling in Groups) Let. Suppose is a -ary - of size . Suppose

further that for each group , there exists acode of size . Then, is ancode of size .

Example 2.5: Take the ternary 4- of type andsize 1600 from Example 2.4. We have an optimalcode by Example 2.1. Fill in all the groups with this optimalcode. The result is an optimal code.

Remark: There is an obvious generalization of Construction2.2 that allows filling in groups with small GDCs to obtainGDCs of larger size. For example, if we fill in all the groups ofthe ternary 4- of type in Example 2.5 with ternary4- s of type and size 120, the result is a ternary4- of type and size 2080.

Construction 2.3: (Inflation) Let . Supposeis a -ary - of size . Let

and . Suppose further that there is a, ,

for any codeword . For each , form a codewordwith for and zero else-

where. Then, is a -ary - oftype and size .

Example 2.6: There is a by Lemma 2.4. Take theternary 4- of type and size 144 from Lemma 2.6.By Inflation, we obtain a ternary 4- of type andsize 2304. We also call this procedure inflating the ternary4- of type by 4.

Construction 2.4 ([6]): (Adjoining Points) Let .Suppose is a -ary - of size . Let be aset of size disjoint from and . Suppose furtherthat

1) for a fixed group , there exists acode of size ;

2) for any group , there exists a -ary- of type

and size .Then, is an

code of size .


TABLE IITYPE AND SOURCE OF STARTING GDCS IN LEMMA 5.2

TABLE IIIBASE CODEWORDS OF SMALL GDCS OF TYPE �

Example 2.7: Here is an optimal code, which isobtained by developing the elements of in the following sup-ports of codewords with the subscripts fixed:

Take the ternary 4- of type and size 2304 fromExample 2.6. Adjoin six ideal points. Fill in the first three groupstogether with the six extra points with ternary 4- s oftype and size 276 from Example 2.3, and fill in the lastgroup together with the six extra points with the aforementionedoptimal code. The result is an optimalcode.

In the sequel of this paper, we only consider ternary CWCswith weight 4 and distance 5, so we simply denote the ternary4- as GDC.

III. SOME SMALL GDCS AND OPTIMAL CODES

In this section, we present some small GDCs and optimalcodes, which are required in establishing subsequent results.

Lemma 3.1: There exists a GDC of type and sizefor each .

Proof: For each , let , and. Then,

is a GDC of type and size , where is obtainedby developing the elements of in the codewords listed inTable III .

Lemma 3.2: There exists a GDC of type and size 224.Proof: Let , and

. Then, is a GDC of type and size 224,where is obtained by developing the elements of in thefollowing codewords :


Proof: For each , let , and. Then,

is a GDC of type and size , where is obtainedby developing the elements of in the following codewords

::

:

:


TABLE IVBASE CODEWORDS OF SOME SMALL GDCS IN LEMMA 3.7

For , inflate GDCs of type (see Example2.1, Lemmas 2.6 and 3.1) by 3 to obtain the required GDCs.


. Then, is a GDC oftype and size 162, where is generated from the followingcodewords, which are developed under the automorphism group

:


Proof: For each , let, and

. Then, is a GDC of typeand size , where is generated from the following

codewords, which are developed under the automorphism group:

:

:

Lemma 3.6: There exists a GDC of type and sizefor all .

Proof: When or and , there existsa -PBD by Lemma 2.3. Deleting one point from


the point set gives a -GDD of type . When orand , there exists a -PBD

by Lemma 2.3. Remove one point not contained in the block ofsize 7 from the point set to obtain a -GDD of type .Hence, we always have a -GDD of type for anyand .

Start from the aforementioned -GDDs of type , andapply the Fundamental Construction with weight 4 to obtainGDCs of type for all and . Here, the inputGDCs of types and exist by Example 2.2 and Lemma 3.2.

For , take a -GDD of type (see [19]), and applythe Fundamental Construction with weight 3 to obtain the re-quired GDC. Here, the input GDC of type is from Lemma2.6.



. Then, is a GDC of typeand size , where is generated from the

codewords listed in Table IV, which are developed under theautomorphism group as indicated next.

For.

For ,.

For ,.


. Then, is a GDC of typeand size 72, where is generated from the following

codewords, which are developed under the automorphism group:

We have the following improvement on for.

Lemma 3.9: ,, and .

Proof: For , the required code is constructed on ,and obtained by developing the elements of in the followingcodewords :

For , the desired code is constructed onwith 48 codewords listed next:

For , take a GDC of type from Lemma 3.6. Adjoinfour ideal points, fill in each of the first three groups togetherwith the ideal points with a GDC of type from Lemma3.8, and fill in the other group together with the ideal pointswith an optimal code to get the desired code.

IV. CASE OF LENGTH

In this section, we focus our attention on the determinationof for . For , we have thefollowing lower bound.

Lemma 4.1: .


TABLE VBASE CODEWORDS OF SMALL OPTIMAL �� CODES FOR � � � ��

TABLE VIBASE CODEWORDS OF GDC OF TYPE � IN LEMMA 4.3


Proof: The desired code is constructed onwith 41 codewords listed next:

Lemma 4.2: for each, , or .

Proof: For , the required code is constructed inExample 2.7. For the other lengths , the required codes areconstructed on , and obtained by developing the elements of

in the codewords listed in Table V .

Lemma 4.3: There exists a GDC of type and size2211.

Proof: Let , and. Then is a GDC of type

and size 2211, where is generated from the codewordslisted in Table VI, which are developed under the automorphismgroup

.

Lemma 4.4: There exists a GDC of type and sizefor all .

Proof: For and , the proof is similar to that ofLemma 3.6. Here, the input GDCs of types and are fromLemmas 2.6 and 3.3.

For , inflate a GDC of type (see Lemma 3.3) by 3 toobtain the required GDC.

Corollary 4.1: for all .Proof: Take GDCs of type from Lemma 4.4 and fill in

all the groups with optimal codes from Lemma 4.2.The results are optimal codes for all .

Lemma 4.5: for each.

Proof: For , take the GDC of type in Lemma4.3 and fill in the group of size 18 with an optimalcode to get the desired code.

For , the desired code is constructed in Example 2.7.

Lemma 4.6: There exist GDCs of type and sizefor all with .

Proof: Take a from Lemma 2.4, and apply theFundamental Construction with weight 6 to all points in the firstfour groups, points in the fifth group, and points in thelast group, where or , . Theother points are given weight 0. The input GDCs of types ,

, and are from Lemmas 2.6 and 3.3. The result is a GDCof type . Adjoin 18 ideal points, and fill inthe first five groups together with the ideal points with GDCs oftypes or to get a GDC of type .The result is a GDC of type with , where

can take any integer no less than 31.

Corollary 4.2: for allwith .

Proof: Take the GDCs of types for all withfrom Lemma 4.6. Fill in the groups of the GDCs

with suitable codes of small lengths from Lemma 4.2 to obtainthe desired codes.

Lemma 4.7: There exist GDCs of type and sizefor each , ,

with .Proof: Take a from Lemma 2.4 and apply the

Fundamental Construction with weight 6 to all points in the firstfour groups, points in the last group, and points in the thremaining group for , and weight 0 to all the otherpoints to get a GDC of type for

, . The input GDCs of typefor are from Lemmas 2.6 and 3.3. Adjoin 18 idealpoints, and fill in the groups together with the ideal points withGDCs of type except for the group of size to get a GDC oftype for , .The result is a GDC of type for ,

, with .

Corollary 4.3: for eachwith .

Proof: Take the GDCs of types for each, , with from

Lemma 4.7. Fill in the groups of the GDCs with suitable codesof small lengths from Lemma 4.2 to obtain the desired codes.

Lemma 4.8: The following GDCs all exist:1) type and size for ;2) type and size for ;3) type and size for ;4) type and size for ;5) type and size 14592;6) type and size 8400.

Proof: The required GDCs of types forare obtained by inflating GDCs of types (see Lemmas 2.6 and3.3) by 4.

The GDCs of types for are obtained byinflating GDCs of types (see Lemmas 2.6 and 3.3) by 5.

The GDCs of types for are ob-tained by applying the Fundamental Construction with weight4 to -GDDs of type (see [20, Th. 1.6]).

For GDCs of types for , take afrom Lemma 2.4, and remove a point to get a -GDD oftype . Apply the Fundamental Construction with


TABLE VIIBASE CODEWORDS OF SMALL OPTIMAL �� CODES FOR � � � ��

weight 6 to all the points in the groups of size 4 and three pointsin the group of size , and weight 0 to the remaining points.The result is a GDC of type .

For a GDC of type , take a and remove apoint to get a -GDD of type . Apply the FundamentalConstruction with weight 6 to all the points in the groups ofsize 4 and five points in the group of size 7, and weight 0 to theremaining points to obtain a GDC of type .

For a GDC of type , take a , apply the Fun-damental Construction with weight 6 to all the points in the firstfive groups, 4 points in the last group, and weight 0 to the re-maining points to obtain a GDC of type .

Corollary 4.4: for eachwith or .

Proof: Take the GDCs constructed in Lemma 4.8. Fill inthe groups of the GDCs with suitable codes of small lengthsfrom Lemma 4.2 to obtain the desired codes.

Summarizing Lemmas 4.1, 4.2, and 4.5, and Corollaries 4.1– 4.4, we obtain the main result of this section.

Theorem 4.1: for all integer, ; .

V. CASE OF LENGTH

In this section, we will determine the value of forall . It is easy to prove that if there exists aGDC of type and size for , thenwe have an optimal code.

Lemma 5.1: There exists a GDC of type and sizefor each , , or .

Proof: For , the requiredGDCs are from Examples 2.1 and 2.5, and Lemmas 2.6 and 3.1.

For each , take a GDC of type for(see Lemma 3.6). Adjoin two ideal points and fill in

the groups together with the ideal points with GDCs of typeto obtain a GDC of type .

For each , take a GDC of type for(see Lemma 3.7), adjoin two ideal points, and fill in the

groups together with the ideal points with GDCs of types orto obtain the required GDC.

Lemma 5.2: There exists a GDC of type and sizefor each , or .

Proof: Take the GDCs of types forand constructed in Lemmas


TABLE VIIIBASE CODEWORDS OF GDCS IN LEMMA 6.3

TABLE IXBASE CODEWORDS OF GDC OF TYPE � �� IN LEMMA 7.2

4.4, 4.6 – 4.8, adjoin two ideal points, and fill in the groupsof size or together with the ideal points with GDCs oftypes or which are from Lemma 5.1. The resultsare GDCs of type . For each desired ,we list the type and source of the starting GDC in Table II.

Combining Lemmas 5.1 and 5.2, we have the following re-sult.

Theorem 5.1: for all integer, .

VI. CASE OF LENGTH

In this section, we determine the value of for all.

Lemma 6.1: .Proof: The desired code is constructed on

with 33 codewords listed next:

Lemma 6.2: for each, , or .

Proof: The required codes are constructed on , and ob-tained by developing the elements of in the codewords listedin Table VII .



. Then, is a GDC of type andsize , where is generated from the codewords listed inTable VIII, which are developed under the automorphism group

.

Lemma 6.4: for andall .

Proof: For , take a from Lemma 2.4, andapply the Fundamental Construction by giving weight 6 to allthe points in the first four groups and one point in the last groupand weight 0 to the remaining points to get a GDC of type .Adjoin five ideal points, fill in the groups of size 24 together withthe extra points with a GDC of type , and fill in the groupof size 6 together with the extra points with an optimal code oflength 11 to get the desired code.

For or , the proof is similar to that ofLemma 5.2. Take the GDCs of types forand constructed in Lemmas


TABLE XBASE CODEWORDS OF SOME SMALL GDCS IN LEMMA 7.3

4.4, 4.6 – 4.8. Adjoin five ideal points, fill in the groups of sizetogether with the ideal points with GDCs of type from

Lemma 6.3, and fill in the group of size together with theideal points with optimal codes of length from Lemma6.2. The results are optimal codes of length .

Combining Lemmas 6.1, 6.2, and 6.4, we have the followingresult.

Theorem 6.1: for all integer, .

VII. CASE OF LENGTH

In this section, we determine the value of for.

Lemma 7.1: .Proof: The required code is constructed on with 67

codewords, which are composed of two parts. The first part con-tains the two codewords and .The other part consists of 65 codewords as follows.

Each of the following 13 codewords will be generated to fivecodewords by applying the following maps for toevery element of a codeword, where


. Then, is a GDC of typeand size 987, where is generated from the codewords

listed in Table IX, which are developed under the automorphismgroup

.

Lemma 7.3: There exists a GDC of type and size, and therefore, , for each

.Proof: For each , let

, and. Then, is a GDC of type and

size , where is generated from the codewordslisted in Table X, which are developed under the automorphismgroup as indicated next.

For , .For ,

.

Lemma 7.4: for alland .

Proof: Take a GDC of type from Lemma 3.6, adjointhree ideal points, fill in the first groups together with theideal points with GDCs of type , and fill in the last grouptogether with the ideal points with an optimal codeto obtain the required codes.

Lemma 7.5: for alland .

Proof: For , take a GDC of type fromLemma 7.2, and fill in the group of size 15 with an optimal

code to get the desired code.


For each , take a GDC of typefor from Lemma 3.7. Adjoin three ideal

points. Fill in one group of size 12 together with the ideal pointswith an optimal code, and fill in the other groupstogether with the ideal points with GDCs of types orto get an optimal code.

For , take a -GDD of type (see [18]), andapply the Fundamental Construction with weight 6 to all thepoints to get a GDC of type . Adjoin three ideal points,fill in one group of size 12 together with the extra points withan optimal code, and fill in the other groups togetherwith the extra points with GDCs of types or to getthe desired code.

For , take a -GDD of type (see [16]),and apply the Fundamental Construction with weight 6 to allthe points to get a GDC of type . Adjoin three idealpoints, fill in the group of size 48 together with the three extrapoints with an optimal code, and fill in the othergroups together with the extra points with GDCs of typesor to get the desired code.

For , take a from Lemma 2.4. Apply theFundamental Construction with weight 3 to all the points in thefirst five groups, weight 3 to 6 points in the sixth group, weight 6to all the points in the last group, and weight 0 to the remainingpoints to get a GDC of type . Here, the input GDCsof types and are from Lemmas 2.6 and 3.4. Adjointhree ideal points, fill in the group of size 48 together with theextra points with an optimal code, and fill in the othergroups together with the extra points with GDCs of typesor to get the desired code.

For or , take a from Lemma2.4. Apply the Fundamental Construction with weight 6 to allpoints in the first four groups, points in the fifth group, twopoints in the last group, and weight 3 to another point in the lastgroup. The other points are given weight 0. The result is a GDCof type for or . Fill in eachgroup of the GDC with suitable optimal codes of lengths , ,or 15 from Theorem 4.1 and Lemma 7.1 to get optimal codes oflength , where can take 23, 25, or any oddinteger no less than 31.

Combining the aforementioned lemmas, we have the fol-lowing theorem.

Theorem 7.1: for and all, ; for each

.

VIII. CONCLUSION

In this paper, we determine almost completely the spectrumof sizes for optimal ternary CWCs of weight 4 and distance 5.We summarize our main results of this paper as follows.

Theorem 8.1: For any integer ,

ifififififif

if and

,

for each , ,and .

ACKNOWLEDGMENT

The authors would like to thank Dr. M. Blaum, the AssociateEditor of the IEEE TRANSACTIONS ON INFORMATION THEORY,and the anonymous referee for their constructive commentsand suggestions that greatly improved the readability of thisarticle. The authors express their gratitude to Prof. L. Ji for thehelpful discussion on the construction method for the optimal

code.

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Hui Zhang is currently a Ph.D. student at Zhejiang University, Hangzhou, Zhe-jiang, China. Her research interests include combinatorial design theory, codingtheory, cryptography, and their interactions.

Xiande Zhang received the Ph.D. degree in mathematics from Zhejiang Uni-versity, Hangzhou, Zhejiang, China, in 2009. During 2009–2011, she held apostdoctoral position with Mathematical Sciences, Nanyang Technological Uni-versity, Singapore. She is now a research fellow with School of MathematicalSciences, Monash University, Australia. Her research interests include combi-natorial design theory, coding theory, cryptography, and their interactions.

Gennian Ge received the M.S. and Ph.D. degrees in mathematics from SuzhouUniversity, Suzhou, Jiangsu, China, in 1993 and 1996, respectively. After that,he became a member of Suzhou University. He was a postdoctoral fellow inthe Department of Computer Science at Concordia University, Montreal, QC,Canada, from September 2001 to August 2002, and a visiting assistant pro-fessor in the Department of Computer Science at the University of Vermont,Burlington, Vermont, USA, from September 2002 to February 2004. Since then,he has been a full professor in the Department of Mathematics at Zhejiang Uni-versity, Hangzhou, Zhejiang, China. His research interests include the construc-tions of combinatorial designs and their applications to codes and crypts.

Dr. Ge is on the Editorial Board of the Journal of Combinatorial Designs, TheOpen Mathematics Journal, and the International Journal of Combinatorics.He received the 2006 Hall Medal from the Institute of Combinatorics and ItsApplications.

Documents

2706 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, …staff.ustc.edu.cn/~drzhangx/papers/ternary cwc.pdf · In Section II, basic notations and results in coding theory and combinatorial