Inequivalent Hadamard Matrices of Order $2n$ Constructed from Hadamard Matrices of Order $n$

Abstract

In this paper, we establish a doubling method to construct inequivalent Hadamard matrices of order $2n$, from Hadamard matrices of order $n$. Our doubling method uses heavily the symmetric group $S_n$, where $n$ is the order of a Hadamard matrix. We improve the efficiency of the method by introducing some group-theoretical heuristics. Using the doubling method in conjunction with the standard 4-row profile criterion, we have constructed several millions of new inequivalent Hadamard matrices of orders 48, 56, 64, 72, 80, 88, 96, and several hundreds of inequivalent Hadamard matrices of orders 672 and 856. The Magma code segments, included in this paper, allow one to compute many more inequivalent Hadamard matrices of the above orders and all other orders of the form $8t$.