Binary Periodic Synchronizing Sequences

In this article, we consider words over f0; 1g of length l 2. The autodistance of such a word is the lowest among the Hamming distances between the word and its images by circular permutations other than identity; the word’s reverse autodistance is the highest among these distances. For each l 2, we study the words of length l whose autodistance and reverse autodistance are close to l=2 (we call such words synchronizing sequences). We establish, for every l 3, an upper bound on the autodistance of words of length l. This upper bound, called up(l), is very close to l=2. We briefly describe the maximal period linear recurring sequences, a previously known family of words over f0; 1g achieving the upper bound up(l) for l = 2 1. Examples of words whose autodistance and reverse autodistance are both equal or close to up(l) are discussed; we describe the method (based on simulated annealing) which permitted the examples to be found. We prove that, for l large enough, an arbitrary majority of words of length l has both its autodistance and its reverse autodistance very close to up(l).