On the Classifications of Weighing Matrices of Order $12$

Abstract

A weighing matrix $A=A(n,k)$ of order $n$ and weight $k$ is a square matrix of order $n$, with entries $0,\pm 1$ which satisfies $A{A}^T=k{I}_n$. H.C. Chan, C.A. Rodger, and J. Seberry “On inequivalent weighing matrices, $ArsCombinatoria$, $(1986)21-A,299-333$” showed that there were exactly $5$ inequivalent weighing matrices of order $12$ and weight $4$ and exactly $2$ inequivalent matrices of weight $5$. They showed that the weighing matrices of order $12$ and weights $2,3$, and $11$ were unique. Q.M. Husain “On the totality of the solutions for the symmetric block designs: $\lambda =2,k=5$ or $6$,” Sanky$\overline{a}$ $7(1945),204-208$” had shown that the Hadamard matrix of order $12$ (the weighing matrix of weight $12$) is unique. In this paper, we complete the classification of weighing matrices of order $12$ by showing that there are seven inequivalent matrices of weight $6$, three of weight $7$, six of weight $8$, four of weight $9$, and four of weight $10$. These results have considerable implications for inequivalence results for orders greater than 12.