Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers

Abstract

Quasi-orthogonal Hadamard matrices and Mersenne matrices with two and three values of the elements, used in digital data processing, are considered, as well as the basis of error-correcting codes and algorithms for transforming orthogonal images. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The connection between symmetry and antisymmetry of structures of cyclic Hadamard and Mersenne matrices on a orders equal to prime numbers, products of close primes, composite numbers, powers of a prime number is shown. Separately, orders equal to the degrees of the prime number 2 are distinguished, both the orders of Hadamard matrices and the basis of the composite orders of Mersenne matrices of block structures with two element values. It is shown that symmetric Hadamard matrices of cyclic and bicyclic structures, according to the extended Riser boundary, do not exist on orders above 32. Mersenne matrices of composite orders belonging to the sequence of Mersenne numbers 2k ‒ 1 nested in the sequence of orders of the main family of Mersenne matrices 4t ‒ 1 exist in a symmetric and antisymmetric form. For orders equal to the powers of a prime number, Mersenne matrices exist in the form of block-diagonal constructions with three element values. The value of prime power determines the number of blocks along the diagonal of the matrix on which the elements with the third value are located. The cyclic blocks are symmetrical and antisymmetric.