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

A code $C$ in the Hamming graph $\Gamma =H(m,q)$ is $2\text{-neighbour-transitive}$ if $\mathrm{A}\mathrm{u}\mathrm{t}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\Gamma$ 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. Thus, as a corollary of the main result, completely transitive codes satisfying the above conditions are also classified.