Gray Cycles of Maximum Length Related to k-Character Substitutions

TL;DR

This paper investigates the maximum size of finite languages over which Gray cycles exist under k-character substitution relations, providing bounds for various alphabet sizes and word lengths.

Contribution

It introduces the complexity measure λ(n) for Gray cycles under k-character substitutions and computes bounds for all alphabet sizes and word lengths.

Findings

Derived bounds for λ(n) across different alphabet sizes.

Established existence conditions for Gray cycles under k-character substitutions.

Provided a comprehensive analysis of the maximum language size for Gray cycles.

Abstract

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.