Quasi-Hadamard matrix

Abstract

We apply the Hadamard equivalence to all the binary matrices of size m × n and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class and count the number of HR-minimals of size m × n for m ≤ 3. Some properties and constructions of HR-minimals are investigated. HR-minimals with the largest weight on its second row are defined as Quasi-Hadamard matrices, which are very similar to Hadamard matrices in terms of the absolute correlations of pairs of rows, in the sense that they give a set of row vectors with “best possible orthogonality.” We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.