On Circulant $G$-Matrices

Abstract

Let $X_1,X_2,X_3,X_4$ be four type 1 $(1,-1)$ matrices on the same group of order $n$ (odd) with the properties: (i) $(X_i–I{)}^T=-(X_i–I)$, $i=1,2$, (ii) $X_i^T=X_i$, $i=3,4$ and the diagonal elements are positive, (iii) $X_i{X}_j=X_j{X}_i$, and (iv) $X_1{X}_1^T+X_2{X}_2^T+X_3{X}_3^T+X_4{X}_4^T=4n{I}_n$. Call such matrices $G$-matrices. If there exist circulant $G$-matrices of order $n$ it can be easily shown that $4n–2=a^2+b^2$, where $a$ and $b$ are odd integers. It is known that they exist for odd $n\le 27$, except for $n=11,17$ for which orders they can not exist. In this paper we give for the first time all non-equivalent circulant $G$-matrices of odd order $n\le 33$ as well as some new non-equivalent circulant $G$-matrices of order $n=37,41$. We note that no $G$-matrices were previously known for orders 31, 33, 37 and 41. These are presented in tables in the form of the corresponding non-equivalent supplementary difference sets. In the sequel we use $G\$-matricestoconstructsome\text{(}F$-matrices and orthogonal designs.