On Optimal Homogeneous-Metric Codes

TL;DR

This paper extends classical coding bounds to finite chain rings with the homogeneous metric, characterizing maximum distance codes and analyzing bounds like Singleton and Plotkin in this context.

Contribution

It provides a complete characterization of maximum homogeneous distance codes and explores bounds and minimal length codes over finite chain rings.

Findings

MHD codes coincide with lifted MDS codes

Complete characterization of MHD codes including exceptions

Identified minimal length codes for constant homogeneous weight

Abstract

The homogeneous metric can be viewed as a natural extension of the Hamming metric to finite chain rings. It distinguishes between three types of elements: zero, non-zero elements in the socle, and elements outside the socle. Since the Singleton bound is one of the most fundamental and widely studied bounds in classical coding theory, we investigate its analogue for codes over finite chain rings equipped with the homogeneous metric. We provide a complete characterization of Maximum Homogeneous Distance (MHD) codes, showing that MHD codes coincide with lifted MDS codes and are contained within the socle at low rank. Exceptions arise from exceptional MDS codes or single-parity-check codes. We then shift our focus to the Plotkin-type bound in the homogeneous metric and close an existing gap in the theory of constant homogeneous-weight codes by identifying those of minimal length.