On Inequivalent Hadamard Matrices of Order $36$

Abstract

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.