2017/03/01 by Hyun, Jong Yoon, Kim, Hyun Kwang, Park, Jeong Rye
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.00139
Etzion et al. introduced metrics on \mathbbF2n 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 \widetildeH3 to be a 2-perfect code. We also construct some families of such structures for any k ≥ 3. Those families enable us to construct packing or covering codes of radius 2 under certain maps.