On maximal almost balanced non-overlapping codes and non-overlapping codes with restricted run-lengths

TL;DR

This paper characterizes and constructs non-overlapping codes with restrictions on symbol balance and run-lengths, improving bounds and providing exact sizes for certain cases relevant to communication and DNA storage.

Contribution

It introduces necessary and sufficient conditions for maximal balanced non-overlapping codes and develops constructions for codes with run-length constraints, addressing practical physical limitations.

Findings

Derived bounds and exact sizes for polarity-balanced non-overlapping codes.

Provided constructions for codes satisfying both balance and run-length constraints.

Improved known bounds on maximum sizes of constrained non-overlapping codes.

Abstract

This paper concerns non-overlapping codes, block codes motivated by synchronisation and DNA-based storage applications. Most existing constructions of these codes do not account for the restrictions posed by the physical properties of communication channels. If undesired sequences are not avoided, the system using the encoding may start behaving incorrectly. Hence, we aim to characterise all non-overlapping codes satisfying two additional constraints. For the first constraint, where approximately half of the letters in each word are positive, we derive necessary and sufficient conditions for the code's non-expandability and improve known bounds on its maximum size. We also determine exact values for the maximum sizes of polarity-balanced non-overlapping codes having small block and alphabet sizes. For the other constraint, where long sequences of consecutive equal symbols lead to undesired behaviour, we derive bounds and constructions of constrained non-overlapping codes. Moreover, we provide constructions of non-overlapping codes that satisfy both constraints and analyse the sizes of the obtained codes.