2-Neighbour-Transitive Codes with Small Blocks of Imprimitivity - Gillespie, Neil I.; Hawtin, Daniel R.; Praeger, Cheryl E. (The Electronic Journal of Combinatorics, 2020)
A code C in the Hamming graph =H(m,q) is 2-neighbour-transitive if Aut(C) acts transitively on each of C=C_0, C_1 and C_2, the first three parts of the distance partition of V with respect to C. Previous classifications of families of 2-neighbour-transitive codes leave only those with an affine action on the alphabet to be investigated. Here, 2-neighbour-transitive codes with minimum distance at least 5 and that contain ``small’’ subcodes as blocks of imprimitivity are classified. When considering codes with minimum distance at least 5, completely transitive codes are a proper subclass of 2-neighbour-transitive codes. This leads, as a corollary of the main result, to a solution of a problem posed by Giudici in 1998 on completely transitive codes.