Sporadic neighbour-transitive codes in Johnson graphs

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.