Group actions on codes in graphs
Daniel R. Hawtin, Cheryl E. Praeger
TL;DR
This chapter surveys the role of group actions in the theory of codes embedded in graphs, focusing on neighbour-transitive and completely transitive codes within distance-regular structures. It develops a unified framework using automorphism groups, distance partitions, and quotient graphs to connect neighbourhood transitivity with broader distance-transitivity phenomena, and introduces invariant-type tools for structural analysis. The work provides both survey and new results across several graph families, including Hamming, Johnson, Kneser, and their $q$-analogues, as well as incidence graphs of generalized quadrangles, with key classifications in the binary Hamming case and nonexistence results that constrain the landscape of highly symmetric codes. It also presents constructions from permutation modules and projective/reed-muller-type codes, linking coding theory to finite geometry and group theory, and highlights numerous open problems guiding future research in symmetric coding theory and incidence-geometry based codes.
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.