The PBD-Closure of Constant-Composition Codes

TL;DR

This paper establishes a PBD-closure property for constant-composition codes, enabling the determination of optimal code sizes for infinite parameter families from single examples, and solves a longstanding problem for certain code parameters.

Contribution

It introduces a PBD-closure result for constant-composition codes and applies it to find optimal code sizes for large lengths, including previously unresolved cases.

Findings

Determined the size of optimal codes with distance four and weight three for all large lengths.

Established PBD-closure for constant-composition codes.

Solved the problem for odd lengths beyond seven and eleven.

Abstract

We show an interesting PBD-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes can be determined for infinite families of parameter sets from just a single example of an optimal code. As an application, the size of several infinite families of optimal constant-composition codes are derived. In particular, the problem of determining the size of optimal constant-composition codes having distance four and weight three is solved for all lengths sufficiently large. This problem was previously unresolved for odd lengths, except for lengths seven and eleven.