Constructing error-correcting binary codes using transitive permutation groups

TL;DR

This paper introduces new lower bounds for binary error-correcting codes by systematically exploring transitive permutation groups, significantly advancing code size estimates for various parameters.

Contribution

It presents novel lower bounds for binary codes using a systematic computer search over transitive permutation groups, a new approach in code construction.

Findings

Established new lower bounds for A_2(n,d) for multiple (n,d) pairs.

Demonstrated the effectiveness of permutation group-based search in code construction.

Provided data supporting the potential of group-theoretic methods in coding theory.

Abstract

Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$, $A_2(24,4) \ge 327680$, $A_2(24,10) \ge 136$, and $A_2(25,6) \ge 17920$. The new lower bounds are a result of a systematic computer search over transitive permutation groups.