Weighted Posets and Digraphs Admitting the Extended Hamming Code to be a Perfect Code

Abstract

Recently, Etzion et al. introduced metrics on 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 the weighted poset metrics and the 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 (H) over tilde (3) to be a two-perfect code. We also construct some families of such structures for any k >= 3, which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps.