Given a word binary relation $\tau$ we define a $\tau$-Gray cycle over a finite language $X$ to be a permutation $\left(w_{[i]}\right)_{0\le i\le |X|-1}$ of $X$ such that each word $w_i$ is an image of the previous word $w_{i-1}$ 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 $n$.

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