On Equivalence of Hadamard Matrices and Projection Properties

Abstract

The classification of Hadamard matrices of orders \(n \geq 32\) remains an open and difficult problem. The definition of equivalent Hadamard matrices gets increasingly complex as \(n\) grows larger. One efficient criterion (\(K\)-boxes) has been used for the construction of inequivalent Hadamard matrices in order \(28\).

In this paper, we use inequivalent projections of Hadamard matrices and their symmetric Hamming distances to check for inequivalent Hadamard matrices. Using this criterion, we have developed two algorithms. The first one achieves finding all inequivalent projections in \(k\) columns as well as classifying Hadamard matrices, and the second, which is faster than the first, uses the symmetric Hamming distance distribution of projections to classify Hadamard matrices. As an example, we apply the second algorithm to the known inequivalent Hadamard matrices of orders \(n = 4, 8, 12, 16, 20, 24\), and \(28\).