Automated Discovery of Improved Constant Weight Binary Codes

TL;DR

This paper introduces automated methods to discover larger constant weight binary codes, improving lower bounds for various parameters using tabu search and a novel greedy heuristic.

Contribution

It presents two automated strategies, tabu search and a new greedy heuristic, for constructing larger constant weight binary codes, advancing bounds for multiple parameters.

Findings

Improved lower bounds for 24 parameter sets.

Two automated strategies successfully constructed larger codes.

The methods outperform previous approaches in certain cases.

Abstract

A constant weight binary code consists of $n$-bit binary codewords, each with exactly $w$ bits equal to 1, such that any two codewords are at least Hamming distance $d$ apart. $A(n,d,w)$ is the maximum size of a constant weight binary code with parameters $n,d,w$. We establish improved lower bounds on $A(n,d,w)$ by constructing new larger codes, for 24 values of $(n,d,w)$ with $6 \leq d \leq 18$ and $18 \leq n \leq 35$. The improved lower bounds come from two strategies. The first is a tabu search that operates at the level of bit swaps. The second is a novel greedy heuristic that repeatedly chooses the candidate codeword that maximizes a randomly-scored histogram of distances to previously-added codewords. These strategies were proposed by CPro1, an automated protocol that generates, implements, and tests diverse strategies for combinatorial constructions.