Energy-minimizing error-correcting codes

TL;DR

This paper demonstrates that many well-known error-correcting codes are universally optimal for minimizing potential energies, explaining their robustness across different channel models, and develops a framework for analyzing such optimality using linear programming bounds.

Contribution

It establishes the universal optimality of several classical codes and links this property to their binomial moments, extending previous mathematical physics analogies and bounds analysis.

Findings

Hamming, Golay, and Reed-Solomon codes are universally optimal.

Universal optimality is linked to minimal binomial moments.

Universal optimality persists even after removing a codeword.

Abstract

We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potential energies when compared with all other codes of the same size and block length. Examples of these universally optimal codes include Hamming, Golay, and Reed-Solomon codes, among many others, and this helps explain their robustness as the channel model varies. Universal optimality of these codes is equivalent to minimality of their binomial moments, which has been proved in many cases by Ashikhmin and Barg. We highlight connections with mathematical physics and the analogy between these results and previous work by Cohn and Kumar in the continuous setting, and we develop a framework for optimizing the linear programming bounds. Furthermore, we show that if these bounds prove a code is universally optimal, then the code remains universally optimal even if one codeword is removed.