Perfect t-codes in Cayley graphs of groups

Abstract

A perfect [Formula: see text]-code in a graph [Formula: see text] is a subset [Formula: see text] of [Formula: see text] such that every vertex of [Formula: see text] is at a distance not more than [Formula: see text], to exactly one vertex of [Formula: see text]. In this paper, we present a new family of perfect [Formula: see text]-codes in Cayley graphs of groups. We proposed the role of the subgroups of a group to create perfect [Formula: see text]-codes by restricting the elements of the left transversal of the subgroups in the given group. Also, we introduce a new decoding algorithm for the all of perfect [Formula: see text]-codes in Cayley graphs. These codes are able to correct every [Formula: see text]-error pattern.