A modular absolute bound condition for primitive association schemes

TL;DR

This paper introduces new upper bounds for the size of primitive association schemes based on algebraic invariants, extending beyond the classical bounds that depend on the scheme's valency.

Contribution

It provides novel upper bounds for primitive association schemes that rely on algebraic invariants of their adjacency algebra, independent of the scheme's valency.

Findings

New bounds depend on invariants of the adjacency algebra

Bounds are applicable to arbitrary primitive schemes

Extends classical absolute bound condition

Abstract

The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field.