Gray Codes for A-Free Strings

  • Matthew B. Squire

Abstract

For any $q \geq 2$, let $\Sigma_{q}=\{0,\ldots,q\!-\!1\}$, and fix a string $A$ over $\Sigma_{q}$. The $A$-free strings of length $n$ are the strings in $\Sigma_{q}^n$ which do not contain $A$ as a contiguous substring. In this paper, we investigate the possibility of listing the $A$-free strings of length $n$ so that successive strings differ in only one position, and by $\pm 1$ in that position. Such a listing is a Gray code for the $A$-free strings of length $n$.

We identify those $q$ and $A$ such that, for infinitely many $n \geq 0$, a Gray code for the $A$-free strings of length $n$ is prohibited by a parity problem. Our parity argument uses techniques similar to those of Guibas and Odlyzko (Journal of Combinatorial Theory A 30 (1981) pp. 183–208) who enumerated the $A$-free strings of length $n$. When $q$ is even, we also give the complementary positive result: for those $A$ for which an infinite number of parity problems do not exist, we construct a Gray code for the $A$-free strings of length $n$ for all $n \geq 0$.

Published
1996-02-14
Article Number
R17