Gray Cycles of Maximum Length Related to k-Character Substitutions

Abstract

Given a binary word relation τ onto A * and a finite language X ⊆ A * , a τ -Gray cycle over X consists in a permutation w [i] 0≤i≤|X|−1 of X such that each word w [i] is an image under τ of the previous word w [i−1] . We define the complexity measure λA,τ (n), equal to the largest cardinality of a language X having words of length at most n, and st. some τ -Gray cycle over X exists. The present paper is concerned with τ = σ k , the so-called k-character substitution, st. (u, v) ∈ σ 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 σ k -Gray cycles.