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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) \(X_iX_j = X_jX_i\), and (iv) \(X_1X_1^T + X_2X_2^T + X_3X_3^T + X_4X_4^T = 4nI_n\). 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 \(n \leq 27\), 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 \(n \leq 33\) 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$-matrices to construct some \(F\)-matrices and orthogonal designs.