Group Homomorphisms as Error Correcting Codes

We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when G is solvable or H is nilpotent, in terms of the normal subgroup structure of G as well as the prime divisors of |G| and |H|. In particular, we show that in the above case, the distance is independent of the subgroup structure of H. We complement this by showing that, in general, the distance depends on the subgroup structure of H.