Hadamard matrices as a result of Scarpis product without cyclic shifts

Abstract

Introduction: Orthogonal Hadamard matrices consisting of elements 1 and –1 (real number) exist for orders that are multiples of 4. The study considers the product of an orthogonal Hadamard matrix and its core, which is called the Scarpis product, and is similar in meaning to the Kronecker product. Purpose: To show by revealing the symmetries of the block Hadamard matrices that their observance contributes to a product that generalizes the Scarpis method to the nonexistence of a finite field. Results: The study demonstrates that orthogonality is an invariant of the product under discussion, subject to the two conditions: one of the multipliers is inserted into the other one, the sign of the elements of the second multiplier taken into account (the Kronecker product), but with a selective action of the sign on the elements and, most importantly, with the cyclic permutation of the core which depends on the insertion location. The paper shows that such shifts can be completely avoided by using symmetries that are characteristic of the universal forms of Hadamard matrices. In addition, this technique is common for many varieties of adjustable Kronecker products. Practical relevance: Orthogonal sequences and effective methods for their finding by the theory of finite fields and groups are of direct practical importance for the problems of noiseless coding, video compression and visual masking.