Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes

TL;DR

This paper constructs commutative association schemes from affine resolvable designs and Latin squares over finite fields related to twin prime powers, Fermat primes, and Mersenne primes, revealing new algebraic combinatorial structures.

Contribution

It introduces a novel method to derive commutative association schemes from designs based on twin prime powers and prime-related finite fields, expanding the class of known schemes.

Findings

Derived commutative association schemes from specific finite field constructions

Determined eigenmatrices of the constructed schemes

Connected designs to algebraic structures related to prime powers

Abstract

For prime powers $q$ and $q+\varepsilon$ where $\varepsilon\in\{1,2\}$, an affine resolvable design from $\mathbb{F}_q$ and Latin squares from $\mathbb{F}_{q+\varepsilon}$ yield a set of symmetric designs if $\varepsilon=2$ and a set of symmetric group divisible designs if $\varepsilon=1$. We show that these designs derive commutative association schemes, and determine their eigenmatrices.