Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

Abstract

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order \(n=2^m-1\) for small \(m\,(m \le 22)\), is given. As far as we know, for \(m\! \in \! \{5,7,11,13,17,19 \}\), all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all \(m\) and nonorientable at least for all \(m \le 19\). For any non prime \(m\), the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to \(m\le 17\).