Computational Enumeration of Andr\'e Planes

TL;DR

This paper develops computational methods to enumerate and classify Andre9 planes of various orders, providing complete counts for small cases and efficient estimation techniques for larger ones.

Contribution

It introduces algorithms for enumerating non-isomorphic Andre9 planes and extends enumeration to higher orders where direct computation is challenging.

Findings

Complete enumeration of Andre9 planes up to order 125.

Efficient methods for counting isomorphism classes at larger orders.

Framework applicable to all dimensions over the kernel.

Abstract

In this paper, we address computational questions surrounding the enumeration of non-isomorphic Andr\'e planes for any prime power order. We are particularly focused on providing a complete enumeration of all such planes for relatively small orders (up to 125), as well as developing computationally efficient ways to count the number of isomorphism classes for other orders where enumeration is infeasible. Andr\'e planes of all dimensions over their kernel are considered.