  The main result here is a characterisation of binary $2$-neighbour-transitive
codes with minimum distance at least $5$ via their minimal subcodes, which are
found to be generated by certain designs. The motivation for studying this
class of codes comes primarily from their relationship to the class of
completely regular codes. The results contained here yield many more examples
of $2$-neighbour-transitive codes than previous classification results of
families of $2$-neighbour-transitive codes. In the process, new lower bounds on
the minimum distance of particular sub-families are produced. Several results
on the structure of $2$-neighbour-transitive codes with arbitrary alphabet size
are also proved. The proofs of the main results apply the classification of
minimal and pre-minimal submodules of the permutation modules over
$\mathbb{F}_2$ for finite $2$-transitive permutation groups.
