A Note on Subgroup Perfect Codes in Cayley Graphs

TL;DR

This paper establishes a precise criterion for when subgroups form perfect codes in finite groups and applies this to classify such codes in extraspecial 2-groups and related groups.

Contribution

It provides a necessary and sufficient condition for subgroup perfect codes in finite groups and classifies them in specific classes of groups, advancing understanding of algebraic coding theory.

Findings

Characterization of subgroup perfect codes in finite groups

Complete classification in extraspecial 2-groups

Identification of perfect codes in groups with extraspecial Sylow 2-subgroups

Abstract

In this paper, we give a necessary and sufficient condition for a subgroup to be a perfect code for finite groups. As an application, we determine all subgroup perfect codes of extraspecial 2-groups and finite groups whose Sylow 2-subgroup is extraspecial.