2016/09/07 by Gillespie, Neil I., Hawtin, Daniel R.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1609.01886
Let X be a subgroup of the full automorphism group of the Hamming graph H(m,q), and C a subset of the vertices of the Hamming graph. We say that C is an (X,2)-neighbour transitive code if X is transitive on C, as well as C1 and C2, the sets of vertices which are distance 1 and 2 from the code. This paper begins the classification of (X,2)-neighbour transitive codes where the action of X on the entries of the Hamming graph has a non-trivial kernel. There exists a subgroup of X with a 2-transitive action on the alphabet; this action is thus almost-simple or affine. If this 2-transitive action is almost simple we say C is alphabet-almost-simple. The main result in this paper states that the only alphabet-almost-simple (X,2)-neighbour transitive code with minimum distance δ≥ 3 is the repetition code in H(3,q), where q≥ 5.