Constructions and bounds for codes with restricted overlaps

TL;DR

This paper explores codes with restricted overlaps, establishing bounds and constructions for binary codes, and offers an elementary proof of a classical lower bound on non-overlapping codes.

Contribution

It introduces new bounds and constructions for codes with specific overlap restrictions and provides an alternative proof of a known lower bound.

Findings

Established general bounds for codes with restricted overlaps

Developed several binary code constructions

Provided an elementary proof of Levenshtein's lower bound

Abstract

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein in 1964.