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