Optimal Ternary Constant-Composition Codes of Weight Four and Distance Five

3742 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 6, JUNE 2011

Optimal Ternary Constant-Composition Codesof Weight Four and Distance Five

Fei Gao and Gennian Ge

Abstract—Constant-composition codes (CCCs) are a generaliza-tion of constant weight codes and permutation codes. The conceptof group divisible codes, an analog of group divisible designs incombinatorial design theory, was first introduced by Chee et al.This new class of codes has been shown to be useful in recursiveconstructions of constant-weight codes and constant-compositioncodes. In this paper, we consider the problem of determining themaximal sizes of ternary constant-composition codes of weight fourand distance five using group divisible codes as the main tools. Wedetermine the exact values for these parameters. The previouslyknown results are those with code length no greater than 10.

Index Terms—Constant-composition codes, constant-weightcodes, group divisible codes, ternary codes.

I. INTRODUCTION

O NE generalization of constant-weight binary codes as weenlarge the alphabet from to ( ) is the concept

of constant-composition codes. The class of constant-composi-tion codes includes the important permutation codes [38] andhas attracted recent interest due to their numerous applications,such as in determining the zero error decision feedback capacityof discrete memoryless channels [45], multiple access commu-nications [23], spherical codes for modulation [24], DNA codes[32], [35], powerline communication [12], [15], and frequencyhopping [14].

While constant-composition codes have been used since theearly 1980s to bound error and erasure probabilities in deci-sion feedback channels [16], their systematic study only beganin late 1990s [4], [5], [41]. Today, the problem of determiningthe maximum size of a constant-composition code constitutesa central problem in their investigation [3], [8]–[10], [12]–[15],[17]–[22], [31], [33], [34], [43], [44], [46]–[48].

In the paper of Svanström et al. [44], some methods for pro-viding upper and lower bounds on the maximum cardinality

Manuscript received August 18, 2010; revised December 03, 2010; acceptedDecember 04, 2010. Date of current version May 25, 2011. G. Ge was supportedin part by the National Outstanding Youth Science Foundation of China underGrant 10825103, in part by the National Natural Science Foundation of Chinaunder Grant 10771193, in part by the Specialized Research Fund for the Doc-toral Program of Higher Education, in part by the Program for New CenturyExcellent Talents in University, and in part by the Zhejiang Provincial NaturalScience Foundation of China under Grant D7080064.

The authors are with the Department of Mathematics, Zhejiang Univer-sity, Hangzhou 310027, Zhejiang, China (e-mail: feigao.chn@gmail.com;gnge@zju.edu.cn).

Communicated by M. Blaum, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2011.2137370

TABLE IVALUES OF � �� AND � �� FOR � � �

of a ternary code with length , minimum Ham-ming distance , and constant composition were presented,and a table of exact values or bounds for in therange was also given there. Here we list the exact valuesof and for codes with length nogreater than 10 in Table I.

The concept of group divisible codes, an analog of group di-visible designs in combinatorial design theory [26], was first in-troduced by Chee et al. in [9]. This new class of codes has beenshown to be useful in recursive constructions of constant-weightcodes and constant-composition codes [8], [9], [49].

The size of ternary constant-weight codes was studied in [6],[11], [27], [42] and the maximal size of ternary constant-com-position codes of weight three for all lengths and compositionshas been determined completely in [9]. In this paper, we shalluse group divisible codes as the main tools to determine themaximal size of ternary constant-composition codes of weightfour and minimal Hamming distance five for all lengths. Thearticle is organized as follows. Section II provides some basicdefinitions and results on combinatorial designs and codingtheory. In Sections III and IV, we determine the exact valuesof case by case. A brief conclusion is presented inSection V.

II. PRELIMINARY

A. Definitions and Notations

The set of nonnegative integers and positive integers are de-noted by and , respectively. The ring is denotedby and we use

to denote a partition of . We also use the notationfor multisets.

All sets considered in this paper are finite if not obviouslyinfinite. If and are finite sets, denotes the set of vectorsof length , where each component of a vector hasvalue in and is indexed by an element of , that is,

, and for each . A -ary code of lengthis a set for some of size . The elements of are

called codewords. The Hamming norm or the Hamming weightof a vector is defined as

. The distance induced by this norm is called the Hamming

GAO AND GE: OPTIMAL TERNARY CONSTANT-COMPOSITION CODES OF WEIGHT FOUR AND DISTANCE FIVE 3743

distance, denoted , so that , for ,. The composition of a vector is the tuple

, where . For any twovectors , define their support as

. We write instead of and alsocall the support of .

A code is said to have distance if for alldistinct , . If for every codeword , thenis said to be of (constant) weight . A -ary code has constantcomposition if every codeword in has composition .

A -ary code of length , distance , and constant weight isreferred to as an -code. Moreover, if every codeword isof constant composition , we refer to it as an -code.The maximum size of an -code is denotedand the -codes achieving this size are called optimal.Note that the following operations do not affect distance andweight properties of an -code:

1) reordering the components of ;2) deleting zero components of .

Consequently, throughout this paper, we restrict our attention tothose compositions , where

.Suppose is a codeword of an -code, where

. Let . We can representequivalently as a -tuple , where

Throughout this paper, we shall often represent codewords ofconstant-composition codes in this form. This has the advantageof being more succinct and more flexible in manipulation.

For ternary constant-composition codes with weight four,there are only two possible compositions, namely and

. Therefore, we shall focus on these two cases in theremaining sections.

B. General Bounds

Since the distance between any two distinct codewords of aconstant-composition code of weight is at least two and atmost , and that , , are pairwise disjoint if hasdistance , we have

Lemma 2.1 (Chee et al. [9]):

The following Johnson-type bound has been proven for con-stant-composition codes.

Theorem 2.1 (Svanström et al. [44]):

for any , where

ifif .

As consequences of Lemma 2.1 and Theorem 2.1, we havethe following results.

Corollary 2.1 (Chee et al. [10]):

Corollary 2.2 (Johnson Bounds):

We shall use the notations andin the rest of this paper.

C. Group Divisible Designs

A set system is a pair , where is a finite set of pointsand , whose elements are called blocks. The order of theset system is , the number of points. For a set of nonnegativeintegers , a set system is said to be -uniform if

for all .Let be a set system and be a

partition of into subsets, called groups. The tripleis a group divisible design (GDD) when every 2-subset ofnot contained in any group appears in exactly one block and

for all and . We denote a GDDby -GDD if is -uniform. The type of a

GDD is the multiset . We use the expo-nential notation to describe the type of a GDD: a GDD of type

is a GDD where there are exactly groups of size ,.

Theorem 2.2 (Brouwer et al. [7]): A -GDD of typeexists if and only if and:

1) or and or ; or2) or and ; or3) and or ; or4) .

with the two exceptions of types and , for which -GDDsdo not exist.

Theorem 2.3 (Ge and Rees [29]): There exists a -GDDof type for every and with

except for and except possibly for

.A transversal design is a -GDD of type .

The following result on the existence of transversal design isused without explicit reference throughout the paper.

Theorem 2.4 (Abel et al. [2]): Let denote the set ofpositive integers such that there exists a . Then, wehave:

1) ;2) ;3) ;4) ;5) .GDDs of different types can be obtained from transversal de-

signs by truncating groups or truncating blocks.

Theorem 2.5 (Truncating Groups, See Hanani [30]): Letbe an integer, . Let . Supposethat there exists a . Let be integerssatisfying , . Then there exists a -GDDof type .

Theorem 2.6 (Truncating Blocks): Let be an integer, ,and let . Suppose there exists a . Then thereexists a -GDD of type .

A parallel class in a GDD is a subset suchthat each point is contained in exactly one block of .A resolvable GDD (RGDD) is a GDD in whichcan be partitioned into parallel classes. Combining the works in[28], [37], [40], we have the following results on the existenceof -RGDDs.

Theorem 2.7: The necessary conditions for the existence ofa -RGDD of type , namely, ,and , are also sufficient except for

and possibly excepting:1) and

;2) and ;3) and ;4) and ;5) and ;6) and ;7) ,

D. Pairwise Balanced Designs

A pairwise balanced design of order with block sizes( ) is a -uniform set system of order , suchthat every pair of distinct elements of occurs in exactly oneblock of . Let be a positive integer. We use

to denote a PBD containing a block of size . If ,this indicates that there is only one block of size in the PBD.On the other hand, if , then there is at least one block ofsize in the PBD.

Theorem 2.8 (Rees and Stinson [36]): Awith exists if and only if , and:

1) or and or ; or

2) or and or .

Theorem 2.9 (Abel et al. [1]): A exists forall and .

E. Packings

Let . A - packing is a -uniform setsystem of order such that every -subset of is con-tained in at most blocks. If , then is allowed to containrepeated blocks.

The packing number is the maximum number ofblocks in any - packing. A - packingis optimal if . If , often one writes

Theorem 2.10 (Stinson et al. [39]): If and, or if and ,

then . Otherwise,. Here

F. Group Divisible Codes

Given and , the restriction of to , written, is the vector such that

ifif .

The constriction of to , written , is the vectorsuch that .

A group divisible code (GDC) of distance is a triple, where is a partition of with

cardinality and is a -ary code of length ,such that for all distinct , andfor all , . Elements of are called groups. Wedenote a GDC of distance as - if is ofconstant weight . If we want to emphasize the composition ofthe codewords, we denote the GDC as - when every

has composition . The type of a GDC is themultiset . As in the case of GDDs, the exponentialnotation is used to describe the type of a GDC. The size of aGDC is .

Note that an -code of size is equivalent to a- of type with size .The above notions were first introduced in [9], where several

useful constructions about GDCs were also given.

Proposition 2.1 (Filling in Groups): Let .Suppose there exists a - of typewith size . Suppose further that for each , ,there exists a -code of size , then there exists a

-code of size . In particular,if and , , are of constant composition , thenis also of constant composition .

Proposition 2.2 (Adjoining Points): Let . Supposethere exists a (master) - of type with size

, and suppose the following (ingredients) also exist:

TABLE IICODEWORDS OF A �� -�� OF TYPE � WITH SIZE 180

1) a -code of size .2) a - of type with size for ,3) a - of type with size if .

Then, there exists a -code of size

Furthermore, if the master and ingredient codes are of constantcomposition, then so is the resulting code.

The following inflation construction is simple, but useful.

Proposition 2.3 (Inflation Construction): Suppose there ex-ists a - of type with size . Sup-pose further that there exists a , then there exists a

- of type with size . If theoriginal GDC is of constant composition , then so is the de-rived GDC.

Theorem 2.11 (Fundamental Construction, [9]): Let, be a (master) GDD, and

be a weight function. Suppose that for each , there existsan (ingredient) - of type . Then thereexists a - of type . Further-more, if the ingredient GDCs are of constant composition ,then is also of constant composition .

III. DETERMINING THE VALUE OF

In this section, we shall determine the value offor all positive integers . We first construct

some - s to obtain the optimal -codes.

A. Some - s

Proposition 3.1: There exist both a - of typewith size 8 and a - of type with size 28.

Proof: is a - of type with size8, where is the set of the vectors

is a - of type with size 28,where is the set of the vectors:

Proposition 3.2: There exist both a - of typewith size 30 and a - of type with size 63.

Proof: is a - of type withsize 30, where is the set of cyclic shifts of the vectors:

and .is a - of type with size 63,

where is the set of cyclic shifts of the vectors: ,and .

Proposition 3.3: There exists a - of type withsize 32.

Proof: is a - of type withsize 32, where is the set of cyclic shifts of the vectors:

Proposition 3.4: There exists a - of type forall .

Proof: When or and , thereexists a by Theorem 2.8. Deleting one pointfrom this PBD gives a -GDD of type . When or

and , there exists aby Theorem 2.8. Remove one point from this PBD which is notin the unique block of size 7 to get a -GDD of type .Hence, we always have a -GDD of type for anyand .

Start from the above -GDDs of type and applythe Fundamental Construction with weight 2 to obtain

- s of type for all and . Here,

TABLE IIIBASE CODEWORDS OF �� -��S OF TYPE � �

TABLE IVBASE CODEWORDS OF �� -��S OF TYPE �

the input codes, - s of types and , exist byProposition 3.1.

When , is a - of typewith size 180, where is the set of vectors in Table II.

Proposition 3.5: There exist both a - oftype with size for each and a

- of type with size for each.

Proof: For the types with , let, . Then is

the required - of type with size ,

where is the set of quasi-cyclic shifts with length 6 of thevectors in Table III.

For the types with , let, . Then

is the required - of type withsize , where is the set of quasi-cyclic shifts withlength 6 of the vectors in Table IV.

B. Case of Length

We will determine the value of for alland .

Proposition 3.6: There exists an optimal -codeof size for each .

Proof: The 20 codewords of an optimal-code are

The 48 codewords of an optimal -code are

The 204 codewords of an optimal -code aregiven by the quasi-cyclic shifts with length 12 of the vectors:

The codewords of an optimal -code forare given by the quasi-cyclic shifts with length

6 of the vectors in Table V.

Proposition 3.7: There exists a - of typewith size for all .

Proof: When . Take a -GDD of type (whichexists by Theorem 2.2) and apply Theorem 2.11 with weight2 using - s of type with size 8 (which exist byProposition 3.1) as ingredients. This gives a - oftype with size .

When . Inflate a - of type with size 32(which exists by Proposition 3.3) by 3 using a (whichexists by Theorem 2.4). This gives a - of typewith size 288.

Proposition 3.8: for all.

Proof: When . Fill in the groups of a -of type by -codes with size 20 (which existby Proposition 3.6). This gives a -code withsize which meets the boundin Corollary 2.2. Together with a -code and a

-code in Proposition 3.6, we complete the proof.

Proposition 3.9: An optimal -code existsfor all .

Proof: By Theorem 2.4, there is a for all. After truncating the last two groups as in Theorem 2.5, we

have a -GDD of type for and. Apply Theorem 2.11 with weight 6 to this GDD using- s of types , and (which exist by Propo-

sition 3.4) to obtain a - of type .Fill in the groups of this GDC with optimal codes of lengths

, and 18 (which all exist by Propositions 3.6 and 3.8).The resulting code has length , wherefor all and . As runs over , will coverthe interval . And as runs over 12 to infinity,the intervals will overlap and cover all the in-tegers no less than 49. It is easy to check that the size of eachof these CCCs is actually . Hence, they areoptimal.

Proposition 3.10: An optimal -code existsfor each .

Proof: When . Take a (which ex-ists by Theorem 2.4) and truncate the last five groups to ob-tain a -GDD of type , where

for each . Apply Theorem 2.11 to thisGDD with weight 6 using ingredient - s of types

for (which exist by Proposition 3.4) to obtaina - of type . Filling inthe groups of the resulting GDC by optimal codes of lengths

gives the required optimal CCCs.Apply the above same method by taking a instead

to obtain the optimal codes of length with .

Proposition 3.11: An optimal -code existsfor each .

Proof: When . Apply Theorem 2.11 to a-GDD of type with weight 2 using a -

of type and fill in the groups with optimal CCCs of lengths12 and 18. The result is an optimal CCC of length .

When . Inflating a - of types orby 3 using a and filling in the groups, we obtain thedesired optimal CCC for or 10, respectively.

When . Take a and truncate it to obtain a-GDD of type . Applying Theorem 2.11 to this GDD

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

with weight 6 using - s of types and as inputingredients, we have a - of type . Filling inthe groups with optimal CCCs gives the result.

When . Take a and delete two points ornothing in the last group to obtain a -GDD of typeor . Apply Theorem 2.11 to these GDDs with weight 6 using

- s of types and and fill in the groups withoptimal CCCs. We obtain the required optimal codes.

When . By truncating the last group of a ,we have a -GDD of type for . Apply The-orem 2.11 to the resulting GDD with weight 6 using ingredient

- s of types and to obtain a - oftype . Filling in the groups with optimal CCCs givesthe result.

When . Adding three points to three parallel classes ofa -RGDD of type , which exists by Theorem 2.7, gives a

-GDD of type . Applying Theorem 2.11 to this GDDwith weight 6 using ingredient - s of types and

and filling in the groups of the resulting GDC will give therequired code.

When . By truncating the last two groups of a, we have a -GDD of type . Apply

Theorem 2.11 to this GDD with weight 6 using ingredient- s of types , and to obtain a -

of type . Filling in the groups of this GDC with op-timal CCCs gives the result.

Combining the previous results, we have

Theorem 3.1: for all .

C. Case of Length

Proposition 3.12: There exists an optimal -codewith size for or 19.

Proof: The 26 codewords of an optimal-code are given by the cyclic shifts of the

vectors: and .The 57 codewords of an optimal -code

are given by the cyclic shifts of the vectors: ,and .

Theorem 3.2: forall .

Proof: For , see Proposition 3.12. When, adjoining one infinite point to a - of type

(see Proposition 3.4) and filling in the groups together withthe extra point by optimal -codes gives an optimal

-code for all .

D. Case of Length

Proposition 3.13: There exists an optimal -codewith size for or 20.

Proof: The 28 codewords of an optimal-code are given by the quasi-cyclic shifts

with length 2 of the vectors: , ,and .

The 60 codewords of an optimal -code aregiven by the quasi-cyclic shifts with length 4 of the followingvectors:

Proof: For , the desired codes come from Propo-sition 3.13. When . Take a - of typeas the master GDC. Adjoin two infinite points and fill in thegroups together with the two extra points with - sof type . The result is a - of type . Thesize of this GDC is which equals

. Hence, this GDC is also an optimal-code as required.

E. Case of Length

Proposition 3.14: There exists an optimal -codewith size for each .

Proof: The codewords of an optimal-code are given by the quasi-cyclic shifts with

length 3 of the following vectors::

Proposition 3.15:for all .

Proof: Truncating the last two groups of a for, we get a -GDD of type for

. Applying Theorem 2.11 to this GDD with weight 6 forall points of the five groups of sizes or and with weights 2,4 or 6 for the points of the remaining group of size , we havea - of type , where and

. Adjoining one infinite point to this GDCand filling in the groups together with the extra point by optimalcodes of lengths , and , we obtain optimalCCCs of length for all . As

goes from 23 to infinity, the intervals willoverlap and cover all the integers no less then 94.

Proposition 3.16:for each .

Proof: The proof is similar to that of Proposition 3.15.When . Truncating the last two groups of a ,

we get a -GDD of type for . Ap-plying Theorem 2.11 to this GDD with weight 6 for all pointsof the five groups of sizes 5 or and with weights 2, 4 or6 for the points of the remaining group of size , we havea - of type , where and

. Adjoining one infinite point to this GDC andfilling in the groups together with the extra point by optimalcodes of lengths 31, and , we obtain the optimalCCCs of length for all .

When . Applying the above same method bytaking a or a instead to solve the cases

or respectively. The input ingredientcodes are - s of types for and

and optimal CCCs of lengths .They all exist by Propositions 3.4, 3.5, and 3.14.

Proposition 3.17:for and 17.

Proof: By Proposition 3.2, there exist - s oftypes and . Inflating these GDCs by 5 using a ,we get - s of types and . Filling in thegroups by optimal -codes gives the optimal codesof lengths 75 and 105.

Proof: For each given , thecodewords of an optimal

-code are given by the quasi-cyclic shifts with length3 of the vectors in [25, Table 1].

Combining the above propositions, we have

F. Case of Length

Proposition 3.19: There exists a - of typefor each .

Proof: For each given ,let and

. Let be the set of quasi-cyclic shiftswith length 2 of the vectors in [25, Table 2]. Thenis the required - of type with size .

Proof: When . The 37 codewords of an optimal-code are given in Table VI.

When . Let . By The-orem 2.9, there exists a for each . Deletingone point from this PBD gives a -GDD of order with groupsizes in . Applying Theorem 2.11 to this GDDwith weight 6 using - s of types ( ) gives a

- with order and group sizes .Adjoining four infinite points to this GDC and filling in thegroups together with the extra points by - s of types

( ) will give a - of type .Filling in the group of size 4 with an optimal code of length 4,

TABLE VI37 CODEWORDS OF AN OPTIMAL �� -CODE

we obtain a code of length . It is easy to check that thiscode has size , hence the code is optimal.

G. Case of Length

Lemma 3.1: for all.

Proof: Let be an optimal -code with length. For each codeword in , we construct a 3-set

consisting of the label of positions which the symbol “1” ap-peared in. It is easy to check that the collection of all thesesets forms the block set of a 2- packing. So by Theorem2.10, we have

Proposition 3.20: There exists an optimal -codewith size for each .

Proof: The required codewords are listed in [25, Table 3].

Proposition 3.21:for all .

Proof: Truncate a for to get a-GDD of type with and .

Apply Theorem 2.11 to this GDD with one point in the groupof size getting weight 4 and all the other points gettingweight 6 to obtain - s of types for

. To do this, we need the ingredient GDCs oftypes , , , and which all exist by Propositions3.4 and 3.5. Adjoin one infinite point to each of these GDCsand fill in the groups together with the extra point with optimalCCCs of lengths , and . The resultant CCCsare optimal codes with lengths for .As goes from 23 to infinity, the intervals willoverlap and cover all the integers no less than 93.

Proof: The proof is similar to that of Proposition 3.21.When . Truncate the last group of a to get a

-GDD of type for . Apply Theorem 2.11 tothis GDD with weight 6 using - s of types and

to obtain a - of type . Adjoining fourinfinite points to this GDC and filling in each of the four groupsof size 24 together with the extra points by - s oftype gives a - of type . Ad-joining one infinite point and filling in the groups together withthe extra point by optimal -codes of lengths

and results in an optimal -code for.

When . Truncate a to get a-GDD of type with and .

Apply Theorem 2.11 to this GDD with one point in the groupof size getting weight 4 and all the other points gettingweight 6 to obtain - s of types for

. To do this, we need the ingredient GDCs oftypes , , , and (see Propositions 3.4 and 3.5).Adjoin one infinite point to each of these GDCs and fill in thegroups together with the extra point by optimal codes of lengths

and . The resultant CCCs are optimal codes withlengths for .

When . Applying the same method by taking aor a instead gives the ranges

or , respectively. The ingredients needed are- s of types and for which all

exist by Propositions 3.4 and 3.5.

Proof: For , the codewords of therequired optimal -code consist of two parts.The first part is the codewords of an optimal -codeon the points . And the second part consists ofthe vectors in [25, Table 4] developed under the permutation

.When , let . The code-

words of an optimal -code consist of threeparts. The first part is the codewords of the required optimal

-code on the points . The secondpart consists of the two vectors in the first row in [25, Table 5]developed under the permutation

. The thirdpart consists of the vectors in the remaining rows devel-oped under the permutation

Combing the above propositions, we have:

Theorem 3.6:for all .

IV. DETERMINING THE VALUE OF

In this section, we will determine the values of. The values for were determined by

Svanström et al. in [44].

A. Some - s

Proposition 4.1: There exists a - of typewith size for each .

TABLE VII30 CODEWORDS OF AN OPTIMAL �� -CODE

Proof: is a - of typewith size 42, where is the set of cyclic shifts of vectors

, and .is a - of type with size 72,

where is the set of cyclic shifts of vectors ,, and .

is a - of type with size110, where is the set of cyclic shifts of vectors ,

, , and .

Proof: is a - of type withsize 48, where is the set of quasi-cyclic shifts with length 4 ofthe vectors

For each given , let , . Letbe the set of cyclic shifts of the following vectors respectively.Then is the required GDC.

: , , and.: , , ,and .: , , ,

, and .

Proposition 4.3: There exist both a - of typewith size 64 and a - of type with size

100.Proof: For each given , let

, . Let bethe set of quasi-cyclic shifts with length 2 of the followingvectors respectively. Then is the required GDC.

Proposition 4.4: There exists a - of type withsize 180.

Proof: is a - of type withsize 180, where is the set of cyclic shifts of the vectors

Proposition 4.5: There exists a - of typewith size for or 5.

Proof: is a - of type withsize 192, where is the set of cyclic shifts of the vectors

is a - of type with size 320,where is the set of cyclic shifts of the vectors

Proposition 4.6: There exist - s of typewith size and of type with sizefor each .

Proof: For each given , let, . Let be the

set of quasi-cyclic shifts with length 2 of the vectors in [25,Table 6]. Then is the required GDC.

For each given , let ,. Let be the set of quasi-

cyclic shifts with length 2 of the vectors in [25, Table 7]. Thenis the required GDC.

Proof: Inflating a - of type ( )in Proposition 4.2 by 3 using a gives the result.

B. Case of Length

Proposition 4.8: An optimal -codewith size exists for each

.Proof: The 30 codewords of an optimal

-code are given in Table VII.For each given , the

codewords of an optimal -code aregiven by the cyclic shifts of the following vectors respectively.

: , and .: , and .

: , , and.

: , , ,and .

For each given , the codewordsof an optimal -code are given by the quasi-cyclicshifts with length 2 of the following vectors respectively.

Proposition 4.9: There exists an optimal -codewith or for all .

Proof: By Theorem 2.9, there exists afor all . Deleting one point in this PBD

results in a -GDD of order with group sizes 3, 4 or5. Applying Theorem 2.11 to this GDD with weight 4 using

- s of types , and gives a -with group sizes 12, 16 or 20, named it as .

Filling in the groups of the with optimal CCCs oflengths 12, 16 and 20 gives an optimal -code with

for all .Adjoining an infinite point to the and filling in the

groups together with the extra point by optimal CCCs of lengths13, 17 and 21 results in an optimal code of lengthfor each .

After adjoining two infinite points to the and fillingin the groups together with the extra point by - sof types , and , we have a - of type .It is easy to check that the size of this GDC equals

. Hence, the resulting GDC is also an optimal CCCof length for each .

Proposition 4.10: There exists an optimal -codeof length or for each

.Proof: For , we take - s of types

and . Adjoin ( ) infinite point to these GDCs and fillin the groups together with the extra point to obtain optimalcodes of lengths and , respectively.

When . By Theorem 2.2, there exist -GDDsof types and . Apply Theorem 2.11 to these GDDs withweight 4 using - s of type as inputs to obtain

- s of types and respectively. If we adjoin

( ) infinite point to these GDCs and fill in the groups to-gether with the extra point, we obtain optimal codes of lengths

and respectively.For , applying Theorem 2.11 to a -GDD of typewith weight 4 using - s of type gives a

- of type . Adjoin ( ) infinite point tothis GDC and fill in the groups together with the extra point,we obtain optimal codes of length .

When . Truncate a to obtain a-GDD of type for . Applying Theorem

2.11 to this GDD with weight 4 using - s of typesand as inputs results in a GDC of type . If we

adjoin ( ) infinite point to these GDCs and fill in thegroups together with the extra point, we obtain optimal codesof lengths for .

For , applying Theorem 2.11 to a -GDD of typewith weight 4 using - s of type gives a

- of type . If we adjoin ( ) infinitepoint to this GDC and fill in the groups together with theextra point, we obtain an optimal -code.

When . Truncate a to obtain a -GDDof type . By applying Theorem 2.11 to this GDD withweight 4 using - s of types and , we get a

- of type . If we adjoin ( ) infi-nite point to this GDC and fill in the groups together with theextra point, we obtain an optimal -code.

Proposition 4.11: An optimal -code withsize exists for each .

Proof: When . Adjoining two infinitepoints to a - of type for and fillingin the groups together with the extra points by - sof type , we obtain a - of type . Since thesize of this GDC equals , we have theresult for and .

When or 20. Truncating a gives a-GDD of type for . Applying Theorem

2.11 to this GDD with weight 4 using - s of typesand gives a - of type . Adjoining

two infinite points to this GDC and filling in the groups togetherwith the extra points by GDCs of types and , we obtainthe optimal codes for .

When . Inflating a - of type by 3 usinga gives a - of type . Filling in thegroups with an optimal code of length 18 to obtain an optimalcode of length 90.

Proposition 4.12: for1) or , and ; or2) and .

Proof: For each given , ,the codewords of an optimal -code consist of thequasi-cyclic shifts with length 2 of the vectors in Table VIII.

For each , , the codewordsof an optimal -code consist of the cyclic shiftsof the vectors in Table IX.

For each , , the codewordsof an optimal -code consist of the cyclic shiftsof the vectors in Table X.

TABLE VIIIBASE CODEWORDS OF OPTIMAL �� -CODES

TABLE IXBASE CODEWORDS OF OPTIMAL �� -CODES

Theorem 4.1: for alland .

C. Case of Length

Before we state the following lemma, we need the concept ofleave graph.

Let be a complete digraph of order with thepoint set and the directed edges

. For each codeword of an -code ,we associate it with a subgraph of as ,where provided

. It is easy to show that if (forotherwise, .). The leave graph of is the subgraph

of , where .

Lemma 4.1: for all.

Proof: Let . Assume that there is a-code with size

. The leave graph of , , will have

directed edges. Since all points have both even in-degrees andout-degrees, the only possible structure of the leave graph wouldbe two duplicate directed edges from one point to another point.

This leads to a contradiction to the fact that the leave graphshould be a simple digraph. Hence, the Johnson bound couldnot be reached.

Proof: The required codewords are listed in Table XI.

Proposition 4.14: An optimal -code withsize exists for all .

Proof: By Theorem 2.4, there exists a for all. Truncate the last two groups of this TD to obtain a-GDD of types or , where

. Applying Theorem 2.11 with weight 4 for all pointsof the five groups of sizes or and with weight 2 for thepoints of the remaining group using - s of types

, , and (see Propositions 4.2 and 4.3) as inputingredients gives a - of types or

. Adjoining one infinite point and filling in thegroups together with the extra point by optimal codes of lengths

, , 11 and 15, we obtain the codes of lengthsfor . As goes from 12 to , the intervals

will overlap and cover all integers no less than 96.

TABLE XBASE CODEWORDS OF OPTIMAL �� -CODES

TABLE XITHE CODEWORDS OF OPTIMAL �� -CODES

Proof: When . By Proposition 4.6, thereexist - s of types and for .Adjoining one point to these GDCs and filling in the groups willgive the result of optimal -codes for

.When . Take a and truncate the

last two groups to obtain -GDDs of types forand . Applying Theorem 2.11 to

these GDDs with only one point from the group of size get-ting weight 2 and all the other points getting weight 4 results inGDCs of types , and .The ingredient GDCs are of types , and for

which all exist by Propositions 4.2 and 4.3. Adjoiningone infinite point to each of these GDCs and filling in the groupstogether with the extra point will give the optimal codes oflengths , and asrequired.

When . Applying the above same method to awith gives the result.

Proposition 4.16: An optimal -code with sizeexists for each with .

Proof: For , the 125 codewords of an op-timal -code consist of the following vectorsdeveloped under the permutation

TABLE XIIBASE CODEWORDS OF OPTIMAL �� -CODES

For , the 174 codewords of an optimal-code consist of the quasi-cyclic shifts with length

9 of the following vectors:

For , the 231 codewords of an op-timal -code consist of the fol-lowing vectors developed under the permutation

For each given , the code-words of an optimal -code consist of two parts. Thefirst part is the codewords of an optimal -code onthe points . And the second part con-sists of the vectors in Table XII developed under the permutation

, where , foror

Combining the above propositions, we have

Theorem 4.2:for all .

V. CONCLUSION

In this paper, we use group divisible codes as the main tool toconstruct all optimal ternary constant-composition codes withweight four and distance five. The main results of this paper canbe restated as follows.

Theorem 5.1: For

ififififotherwise.

Theorem 5.2: For

ififififif and

otherwise.

ACKNOWLEDGMENT

The authors would like to thank Prof. M. Blaum, the Asso-ciate Editor, Prof. H. Bölcskei, the Editor-in-Chief of the IEEETRANSACTIONS ON INFORMATION THEORY, and the two anony-mous referees for their constructive comments and suggestionsthat greatly improved the readability of this article.

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Fei Gao is currently pursuing the Ph.D. degree at Zhejiang University,Hangzhou, Zhejiang, China.

His research interests include combinatorial 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 receiving his degrees, he became a member of Suzhou University. Hewas a postdoctoral fellow in the Department of Computer Science at ConcordiaUniversity, Montreal, QC, Canada, from September 2001 to August 2002, and avisiting assistant professor in the Department of Computer Science at the Uni-versity of Vermont, Burlington, VT, from September 2002 to February 2004.Since then, he has been a Full Professor in the Department of Mathematics,Zhejiang University, Hangzhou, Zhejiang, China. His research interests includethe constructions of combinatorial designs and their applications to codes andcrypts.

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.

Optimal Ternary Constant-Composition Codes of Weight Four and Distance Five

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