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 ≤ t ≤ 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 2 k , 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 ≥ 2 k , 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 ≥ 4, and a construction for q -ary (1, k )-overlap-free codes of length k + 2 is given.