  We classify the neighbour-transitive codes in Johnson graphs J(v, k) of
minimum distance at least three which admit a neighbour-transitive group of
automorphisms that is an almost simple two-transitive group of degree v and
does not occur in an infinite family of two-transitive groups. The result of
this classification is a table of 22 codes with these properties. Many have
relatively large minimum distance in comparison to their length v and number of
code words. We construct an additional five neighbour-transitive codes with
minimum distance two admitting such a group. All 27 codes are t-designs with t
at least two.
