Hadamard equivalence of binary matrices

Abstract

In this paper, we propose a fast algorithm for checking the Hadamard equivalence of two binary matrices, and give an intuitive analysis on its time complexity. For this, we define Hadamard-equivalence on the set of binary matrices, and a function which induces a total order on them. With respect to this order relation, we define the minimal element which is used as a representative of an equivalence class. We applied the proposed algorithm to Hadamard matrices of smaller sizes, and show the results. Especially, the result for those of Payley type I and II of the same size 60 shows they are not equivalent. Finally, we discuss a new combinatorial problem of counting the number of and enumerating all the inequivalent binary minimal matrices of size m×n, and show the solutions for small values of m, n ≤ 4, leaving many of the observed properties as open problems.