Unbiased weighing matrices of weight $9$

TL;DR

This paper explores the construction and existence of unbiased weighing matrices of weight 9, identifying the smallest order for such pairs and the maximum size for sets of these matrices.

Contribution

It introduces a construction method using mutually suitable Latin squares and determines the maximum sizes of unbiased weighing matrix sets for orders up to 16.

Findings

13 is the smallest order with unbiased weighing matrix pairs

16 is the first order with a maximum class of such matrices

Provides a new construction method for these matrices

Abstract

We investigate unbiased weighing matrices of weight $9$ and provide a construction method using mutually suitable Latin squares. For $n \le 16$, we determine the maximum size among sets of mutually unbiased weighing matrices of order $n$ and weight $9$. Notably, our findings reveal that $13$ is the smallest order where such pairs exist, and $16$ is the first order for which a maximum class of unbiased weighing matrices is found.