Minimal Binary $2$-Neighbour-Transitive Codes

TL;DR

This paper characterizes binary 2-neighbour-transitive codes with minimum distance at least 5 through their minimal subcodes generated by designs, expanding the known examples and establishing new bounds and structural results.

Contribution

It provides a new characterization of binary 2-neighbour-transitive codes via minimal subcodes and designs, and extends the classification to codes with arbitrary alphabet sizes.

Findings

Many new examples of 2-neighbour-transitive codes identified.

New lower bounds on minimum distance established.

Structural properties of codes with arbitrary alphabet size proved.

Abstract

The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of $2$-neighbour-transitive codes than previous classification results of families of $2$-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of $2$-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over $\mathbb{F}_2$ for finite $2$-transitive permutation groups.