Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

TL;DR

This paper introduces a new construction method for optimal constant-weight and constant-composition codes with specific parameters, solving an open problem for large lengths and many small cases.

Contribution

It provides a novel construction based on generalized difference triangle sets, determining exact sizes of optimal codes for all sufficiently large lengths.

Findings

Optimal codes of weight w and distance 2w-1 are characterized for large lengths.

Exact sizes are determined for all such codes with weight w ≤ 6, except two cases.

The construction generalizes previous methods and solves an open problem.

Abstract

An optimal constant-composition or constant-weight code of weight $w$ has linear size if and only if its distance $d$ is at least $2w-1$. When $d\geq 2w$, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of $d=2w-1$ has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight $w$ and distance $2w-1$ based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight $w$ and distance $2w-1$ are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight $w$ and distance $2w-1$ are also determined for all $w\leq 6$, except in two cases.