Conway groupoids, regular two-graphs and supersimple designs

Twisted permutation codes, introduced recently by the second and third authors, are frequency permutation arrays. They are similar to repetition permutation codes, in that they are obtained by a repetition construction applied to a smaller code. It was previously shown that the minimum distance of a twisted permutation code is at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes.