Group actions on codes in graphs

TL;DR

This chapter explores the symmetry properties of codes in distance regular graphs, introducing concepts like completely transitive and neighbour-transitive codes, with applications to well-known graph families.

Contribution

It provides an overview and original results on codes with symmetries in distance regular graphs, focusing on group actions and transitivity notions.

Findings

Development of the strongest notion of completely transitive codes

Introduction of neighbour-transitive codes

Analysis of codes in Hamming, Johnson, and Kneser graphs

Abstract

This is a chapter in a forthcoming book on completely regular codes in distance regular graphs. The chapter provides an overview, and some original results, on codes in distance regular graphs which admit symmetries via a permutation group acting on the vertices of the graph. The strongest notion of completely transitive codes is developed, as well as the more general notion of neighbour-transitive codes. The graphs considered are the Hamming, Johnson, and Kneser graphs and their q-analogues, as well as some graphs related to incidence structures.