In search of maximum non-overlapping codes

TL;DR

This paper characterizes maximal non-overlapping codes, formulates the maximum size problem as an integer optimization, and demonstrates that known constructions are often suboptimal for many alphabet sizes and codeword lengths.

Contribution

It introduces a new characterization of maximal non-overlapping codes and formulates the maximum size problem as an integer optimization, revealing the suboptimality of existing constructions.

Findings

Maximum non-overlapping code sizes are known only for specific cases.

Integer optimization formulations can identify larger codes than existing methods.

Many known constructions are not optimal for various alphabet sizes and codeword lengths.

Abstract

Non-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.