Constructions and Bounds for q-Ary (1, k)-Overlap-Free Codes

A $(1,k)$ -overlap-free code, motivated by applications in DNA-based data storage systems and synchronization between communication devices, is a set of words in which no prefix of length $t$ of any word is the suffix of any word for every integer $t$ such that $1\leq t\leq k$ . A $(1,n-1)$ -overlap-free code of length $n$ is said to be non-overlapping. We provide a construction for $q$ -ary $(1,k)$ -overlap-free codes of length $2k$ , which can be viewed as a generalization of the Zero Block Construction presented by Blackburn, Esfahani, Kreher and Stinson recently over a binary alphabet, and analyze the asymptotic behavior of their sizes. When $n\geq 2k$ , an explicit general lower bound and an asymptotic lower bound for the size of an optimal $q$ -ary $(1,k)$ -overlap-free code of length $n$ are presented. The exact value of the maximum size of $q$ -ary (1, 2)-overlap-free codes of length $n$ is determined for any $n\geq 4$ , and a construction for $q$ -ary $(1,k)$ -overlap-free codes of length $k+2$ is given.