The problem of classification of Hadamard matrices becomes an NP-hard problem as the order of the Hadamard matrices increases. In this paper, we use a new criterion which inspired us to develop an efficient algorithm to investigate the lower bound of inequivalent Hadamard matrices of order \(36\). Using four \((1,-1)\) circulant matrices of order \(9\) in the Goethals-Seidel array, we obtain many new Hadamard matrices of order \(36\) and we show that there are at least \(1036\) inequivalent Hadamard matrices for this order.
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