Gray Cycles of Maximum Length Related to k-Character Substitutions

Given a word binary relation $\tau$ we define a $\tau$-Gray cycle over a finite language X to be a permutation w [i] 0$\le$i$\le$|X|--1 of X such that each word wi is an image of the previous word wi--1 by $\tau$. In that framework, we introduce the complexity measure $\lambda$(n), equal to the largest cardinality of a language X having words of length at most n, and such that a $\tau$-Gray cycle over X exists. The present paper is concerned with the relation $\tau$ = $\sigma$ k , the so-called k-character substitution, where (u, v) belongs to $\sigma$ k if, and only if, the Hamming distance of u and v is k. We compute the bound $\lambda$(n) for all cases of the alphabet cardinality and the argument n.