Sporadic neighbour-transitive codes in Johnson graphs

TL;DR

This paper classifies sporadic neighbour-transitive codes in Johnson graphs with specific symmetry properties, identifying 27 such codes, many with large minimum distances, and showing they form t-designs.

Contribution

It provides a complete classification of certain sporadic neighbour-transitive codes in Johnson graphs, including 22 classified codes and 5 additional constructions, expanding understanding of their structure.

Findings

Identified 22 codes with specific neighbour-transitivity properties

Constructed 5 additional codes with minimum distance two

All 27 codes are t-designs with t ≥ 2

Abstract

We classify the neighbour-transitive codes in Johnson graphs J(v, k) of minimum distance at least three which admit a neighbour-transitive group of automorphisms that is an almost simple two-transitive group of degree v and does not occur in an infinite family of two-transitive groups. The result of this classification is a table of 22 codes with these properties. Many have relatively large minimum distance in comparison to their length v and number of code words. We construct an additional five neighbour-transitive codes with minimum distance two admitting such a group. All 27 codes are t-designs with t at least two.