The weighted poset metrics and directed graph metrics

TL;DR

This paper explores the relationship between weighted poset metrics and directed graph metrics, classifies certain structures that admit extended Hamming codes as perfect codes, and constructs families of such codes for various parameters.

Contribution

It introduces a classification of weighted posets and directed graphs that admit extended Hamming codes as perfect codes, extending the theory of metric spaces in coding.

Findings

Classified weighted posets on 8 elements admitting extended Hamming codes as 2-perfect codes.

Classified directed graphs on 8 vertices with similar properties.

Constructed families of codes for any k ≥ 3 enabling packing or covering codes of radius 2.

Abstract

Etzion et al. introduced metrics on $\mathbb{F}_2^n$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between weighted poset metrics and directed graph based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices which admit the extended Hamming code $\widetilde{\mathcal{H}}_3$ to be a $2$-perfect code. We also construct some families of such structures for any $k \geq 3$. Those families enable us to construct packing or covering codes of radius 2 under certain maps.