A note on classification of binary signal set in the view of Hadamard equivalence

In this paper, we derive a kind of classification of binary signal set by adopting Hadamard equivalence of binary matrices. We propose a fast algorithm for checking the Hadamard equivalence for general 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 binary matrices of smaller sizes, and show the results. Finally, we discuss a new combinatorial problem of counting the number of and enumerating all the inequivalent binary minimal matrices of size m x n, and show the solutions for small sizes, leaving many of the observed properties as open problems.