Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

Given a binary word relation $\tau$ onto $A^*$ and a finite language $X\subseteq A^*$, a $\tau$-Gray cycle over $X$ consists in a permutation $\left(w_{[i]}\right)_{0\le i\le |X|-1}$ of $X$ such that each word $w_{[i]}$ is an image under $\tau$ of the previous word $w_{{[i-1]}}$. We define the complexity measure $\lambda_{A,\tau}(n)$, equal to the largest cardinality of a language $X$ having words of length at most $n$, and s.t. some $\tau$-Gray cycle over $X$ exists. The present paper is concerned with $\tau=\sigma_k$, the so-called $k$-character substitution, s.t. $(u,v)\in\sigma_k$ holds if, and only if, the Hamming distance of $u$ and $v$ is $k$. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length $$\sigma_k$-Gray cycles.