A Construction of Some Group Codes

Abstract

Let Σ* be the free monoid over a finite alphabet Σ and H a subgroup of a given group G. A group code X is the minimal generator of X* with X* =Ψ^(-1)(H), where Ψ is a morphism from the free monoid Σ* to the group G. In general, it is not obvious to detect if a subset X of Σ* is a code or not. In this paper, we use the fact that the syntactic monoid M(X*) of X* is isomorphic to the transition monoid of the minimal automaton recognizing X*, to giving some examples of groups codes based on the following two results from [1]: 1) The subset X of Σ* is a group code if and only if the monoid M(X*) is a group. 2) Let X □Σ* be a finite code, the syntactic monoid M(X*) is a group if and only if X =Σ^n for some a positif integer n. And in this case, the group M(X*) is a cyclic group of order n.