$2$-Neighbour-Transitive Codes with Small Blocks of Imprimitivity

Abstract

A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2$-neighbour-transitive if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ 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.