On inequivalent Hadamard Matrices of Order $44$

Abstract

In this paper, we construct many Hadamard matrices of order $44$ and we use a new efficient algorithm to investigate the lower bound of inequivalent Hadamard matrices of order $44$. Using four $(1,-1)$-circulant matrices of order $11$ in the Goethals-Seidel array, we obtain many new Hadamard matrices of order $44$ and we show that there are at least $6018$ inequivalent Hadamard matrices for this order. Moreover, we use a known method to investigate the existence of double even self-dual codes $[88,44,d]$ over $\text{GF}(2)$ constructed from these Hadamard matrices.