Gray Codes and Strongly Square-Free Strings

Abstract

A binary Gray code is a circular list of all 2ⁿ binary strings of length n such that there is a single bit change between adjacent strings. The coordinate sequence of a binary Gray code of order n is viewed as a circular string of 2ⁿ integers chosen from {1,..., n} so that adjacent strings of the Gray code differ in that bit determined by the corresponding integer in the coordinate sequence. A deleted coordinate sequence of a binary Gray code is a linear sequence obtained from the coordinate sequence by deletion of exactly one integer.

A string is strongly square-free if it contains no “Abelian square”; i.e., a factor adjacent to a permutation of itself. Every deleted coordinate sequence of a Gray code is a strongly square-free string of length 2ⁿ - 1 over {1,..., n}. We further show that Gray codes equivalent to the reflected binary Gray code are characterized by having only deleted coordinate sequences which are maximal strongly square-free strings.