2024/12/23 by Xuanlong Ma, Ma, Xuanlong, Yuefeng Yang +3
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Cooperative Communication and Network Coding
paper · pdf · doi:10.48550/arxiv.2412.17509
Given a finite group G with identity e and a normal subgroup H of G, the subgroup sum graph ΓG,H (resp. extended subgroup sum graph ΓG,H+) of G with respect to H is the graph with vertex set G, in which distinct vertices x and y are adjacent whenever xy∈ H∖ \e\ (resp. xy∈ H). A group G is said to be \em code-perfect if for any normal subgroup H of G, ΓG,H admits a perfect code. In this paper, we give a necessary and sufficient condition for which normal subgroups H of G satisfy that a (extended) subgroup sum graph of G with respect to H admits a (total) perfect code, and classify all code-perfect Dedekind groups. As an application, we classify all normal subgroups such that the subgroup sum graph of a cyclic group, a dihedral group or a dicyclic group with respect to such a normal subgroup admits perfect codes, respectively. We also determine all abelian groups A and subgroups H of A such that ΓA,H admits a total perfect code.