A family of balanced generalized weighing matrices

TL;DR

This paper constructs a new infinite class of balanced weighing matrices with unique parameters, demonstrating their equivalence to a five-class association scheme, thus expanding the known families of such matrices.

Contribution

It introduces the first infinite family of balanced weighing matrices with non-classical parameters, linking them to association schemes.

Findings

Constructed an infinite family of balanced weighing matrices.

Proved their equivalence to five-class association schemes.

Expanded the known classes of balanced weighing matrices.

Abstract

Balanced weighing matrices with parameters $$ \left(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^m\right), $$ for each nonzero integer $m$ is constructed. This is the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme.