Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

3552 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008 Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three Yeow Meng Chee, Senior Member, IEEE, Gennian Ge, and Alan C. H. Ling Abstract—The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which en- abled the determination of the sizes of optimal constant-composi- tion codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length. Index Terms—Constant-composition codes, group divisible codes, group divisible designs, recursive constructions. I. INTRODUCTION O NE generalization of constant-weight binary codes as we enlarge the alphabet from size two to beyond, is the con- cept of constant-composition codes. The class of constant-com- position codes includes the important permutation codes and have attracted recent interest due to their numerous applications, such as in determining the zero error decision feedback capacity of discrete memoryless channels [1], multiple-access communi- cations [2], spherical codes for modulation [3], DNA codes [4], [5], powerline communications [6], [7], and frequency hopping [8]. While constant-composition codes have been used since the early 1980s to bound error and erasure probabilities in decision feedback channels [9], their systematic study only began in late 1990s with Svanström [10]. Today, the problem of determining the maximum size of a constant-composition code constitutes a central problem in their investigation [6], [7], [11]–[20]. Our interest in this paper is in determining the maximum sizes of constant-composition codes of weight three. The techniques Manuscript received September 20, 2007; revised April 3, 2008. The work of Y. M. Chee was supported in part by the Singapore National Research Foundation, the Singapore Ministry of Education under Research Grant T206B2204 and by the Nanyang Technological University under Research Grant M58110040. The work of G. Ge was supported by in part by the National Natural Science Foundation of China under Grant 10771193, the Zhejiang Provincial Natural Science Foundation of China, and the Program for New Century Excellent Talents in University. This work was done when A. C. H. Ling was on sabbatical leave at the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. Y. M. Chee is with the Division of Mathematical Sciences, School of Phys- ical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore (e-mail: ymchee@ntu.edu.sg). G. Ge is with the Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China (e-mail: gnge@zju.edu.cn). A. C. H. Ling is with the Department of Computer Science, University of Vermont, Burlington, VT 05405 USA (e-mail: aling@emba.uvm.edu). Communicated by T. Etzion, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2008.926349 introduced in this paper are built upon the authors’ earlier re- sults [12], where pairwise balanced designs and group divisible designs are used to obtain optimal constant-composition codes of sufficiently large lengths. We remarked in that paper that the techniques developed therein, together with deeper methods in combinatorial design theory, can be used to derive optimal con- stant-composition codes of all lengths, except for a small fi- nite set. In this paper, we show how this can be done by intro- ducing the concept of group divisible codes and applying it to the problem of determining the sizes of optimal constant-com- position codes of weight three. The power of group divisible codes lies in their similarity to group divisible designs, which allow the use of Wilson-type constructions [21], [22]. We begin by reviewing some coding theoretic terminology and notations. The set of integers is denoted by . The ring is denoted by , and the set of nonnegative integers and positive integers are denote by and , respectively. The notation is used for multisets. All sets considered in this paper are finite if not obviously infinite. If and are finite sets, denotes the set of vectors of length , where each component of a vector has value in and is indexed by an element of , that is, , and for each . A -ary code of length is a set for some of size . The elements of are called codewords. The Hamming norm or the Hamming weight of a vector is defined as . The distance induced by this norm is called the Hamming distance, denoted , so that , for . The composition of a vector is the tuple , where . For any two vectors

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