Construction of infinite families of non-Schurian association schemes of order $2p^2$, $p$ an odd prime, based on biaffine planes and Heisenberg groups: research report and beyond

TL;DR

This paper constructs infinite families of non-Schurian association schemes of order 2p^2 using biaffine planes and Heisenberg groups, expanding the understanding of algebraic combinatorics.

Contribution

It introduces a novel construction method for non-Schurian association schemes based on geometric and algebraic models, applicable for all odd primes p.

Findings

Four non-Schurian schemes for each p ≥ 5

Two non-Schurian schemes for p=3

Construction based on incidences in biaffine planes and Heisenberg groups

Abstract

Let $p$ be an odd prime. In this paper we provide a construction which gives four non-Schurian association schemes for every $p\geq 5$ and two for $p=3$. This construction is explained using incidences between points and lines of a biaffine plane and we also provide a pure algebraic model for it with the aid of finite Heisenberg groups. The obtained results are discussed in a more wide framework.