Hamming graphs in Nomura Algebras

TL;DR

This paper demonstrates that the Bose-Mesner algebra of generalized Hamming schemes derived from association schemes is not the Nomura algebra of any type II matrix, providing new examples of self-dual Bose-Mesner algebras outside this class.

Contribution

It establishes that certain Bose-Mesner algebras from generalized Hamming schemes cannot be realized as Nomura algebras of type II matrices, expanding understanding of algebraic structures in association schemes.

Findings

Bose-Mesner algebra of H(n,A) is not a Nomura algebra of a type II matrix

Provides examples of self-dual Bose-Mesner algebras not arising from type II matrices

Clarifies the relationship between association schemes and type II matrices

Abstract

Let A be an association scheme on q\geq 3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n\geq 2, is not the Nomura algebra of a type II matrix. This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices.