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

TL;DR

This paper introduces group divisible codes, a new class of codes that facilitate the construction of optimal constant-composition codes of weight three, significantly advancing the understanding of their sizes and properties.

Contribution

It defines group divisible codes and demonstrates their application in constructing and determining sizes of optimal constant-composition codes of weight three.

Findings

Constructed large classes of group divisible codes

Determined sizes of optimal constant-composition codes for most cases

Only four cases of code sizes remain undetermined

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 enabled the determination of the sizes of optimal constant-composition 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.