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

TL;DR

This paper classifies all self-embedding monomial power permutations of Hamming Steiner triple systems of small order, revealing new embeddings and their properties, and proving nonexistence results for certain cases.

Contribution

It provides a complete classification of self-embeddings for small orders and introduces new embeddings, especially for specific small m values, with proofs of nonexistence for non-prime m.

Findings

All self-embeddings for m in {5,7,11,13,17,19} are new.

Self-embeddings are cyclic and nonorientable for all considered m.

Nonexistence of such embeddings in closed surfaces for non-prime m.

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 < 23, 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 < 21. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven.