Evolutionary Strategies for the Design of Binary Linear Codes

TL;DR

This paper introduces an evolutionary strategy for designing linear binary error-correcting codes, successfully finding optimal codes for lengths up to 14 and exploring code diversity, with performance decline at larger lengths.

Contribution

The paper presents a novel ES algorithm that specifically explores linear codes, maintaining rank constraints, and demonstrates its effectiveness in finding optimal codes for certain lengths.

Findings

Successful convergence to optimal codes up to length 14

Evolved codes are often inequivalent to known best codes

Performance declines for lengths beyond 14, with codes becoming equivalent to known solutions

Abstract

The design of binary error-correcting codes is a challenging optimization problem with several applications in telecommunications and storage, which has also been addressed with metaheuristic techniques and evolutionary algorithms. Still, all these efforts focused on optimizing the minimum distance of unrestricted binary codes, i.e., with no constraints on their linearity, which is a desirable property for efficient implementations. In this paper, we present an Evolutionary Strategy (ES) algorithm that explores only the subset of linear codes of a fixed length and dimension. To that end, we represent the candidate solutions as binary matrices and devise variation operators that preserve their ranks. Our experiments show that up to length $n=14$, our ES always converges to an optimal solution with a full success rate, and the evolved codes are all inequivalent to the Best-Known Linear Code (BKLC) given by MAGMA. On the other hand, for larger lengths, both the success rate of the ES as well as the diversity of the evolved codes start to drop, with the extreme case of $(16,8,5)$ codes which all turn out to be equivalent to MAGMA's BKLC.