Codes with restricted overlaps: expandability, constructions, and bounds

TL;DR

This paper investigates codes with restricted overlaps, providing bounds on their sizes, new constructions for various overlap restrictions, and characterizing non-expandable codes, advancing understanding of overlap-free code design.

Contribution

It introduces new bounds, constructions, and characterizations for codes with specific overlap restrictions, including expandability and non-expandability properties.

Findings

Established a general upper bound on code size with overlap restrictions.

Constructed new overlap-free codes for various parameters, including $(1,k)$-overlap-free codes.

Characterized non-expandable codes when overlaps of certain lengths are prohibited.

Abstract

Consider a $q$-ary block code satisfying the property that no $l$-letters long codeword's prefix occurs as a suffix of any codeword for $l$ inside some interval. We determine a general upper bound on the maximum size of these codes and a tighter bound for codes where overlaps with lengths not exceeding $k$ are prohibited. We then provide constructions for codes with various restrictions on overlap lengths and use them to determine lower bounds on the maximum sizes. In particular, we construct $(1,k)$-overlap-free codes where $k \geq n/2$ and $n$ denotes the block size, expand a known construction of $(k,n-1)$-overlap-free codes, and combine the ideas behind both constructions to obtain $(t_1,t_2)$-overlap-free codes and codes that are simultaneously $(1,k)$- and $(n-k,n-1)$-overlap-free for some $k < n/2$. In the case when overlaps of lengths between 1 and $k$ are prohibited, we complete the characterisation of non-expandable codes started by Cai, Wang, and Feng (2023).