SEMINAR: Groups and Combinatorics Seminar: 2-Neighbour Transitive Codes in Hamming Graphs

Time and place: 15:00 Friday 7 August in Blakers LT.

Speaker: Daniel Hawtin (University of Western Australia)

Title: 2-Neighbour Transitive Codes in Hamming Graphs

Abstract: We consider a code C to be a subset of the vertex set of a Hamming graph, with alphabet Q and entries M. A 1-neighbour is a vertex which is not contained in C, but is distance 1 from some codeword in C. A 2-neighbour is a vertex of the Hamming graph which is distance 2 from some codeword, but not distance 1 or 0 from any codeword. Let X be a subgroup of the automorphism group of the Hamming graph. We say a code C is (X,2)-neighbour transitive if X is transitive on C, the set of 1-neighbours, and the set of 2-neighbours.

Let X_1 be the subgroup of X which fixes the first entry. Gillespie, Giudici and Praeger showed that the action X_1^Q, of X_1 on the alphabet in the first entry, is 2-transitive, and thus affine or almost-simple. In this talk we first address the case where X acts faithfully on M, and then when X_1^Q is almost-simple.