This article discusses strategies for forming monocyclic, two-level quasi-orthogonal matrices with symmetry, aimed at expanding the class of structured matrices applicable to information processing problems. The relevance of the study is due to the increasing demands on the computational efficiency of modern information and telecommunication systems. The goal of the work is to systematize and expand the set of strategies for forming monocyclic, quasi-orthogonal Mersenne matrices, which are the kernel of Hadamard matrices. The proposed approach is based on establishing a connection between the theory of quasi-orthogonal matrices and the theory of binary code sequences with a two-level periodic autocorrelation function, and the use of cyclic Hadamard difference sets. The paper identifies ten strategies for forming the first rows of matrices, differing in the type of base sequences and covering orders associated with prime numbers, products of twin primes, and Mersenne primes. Using specific examples of 31st- and 63rd-order matrices, the possibility of obtaining both symmetric and persymmetric structures was experimentally demonstrated. The obtained results provide a basis for increasing the diversity of quasi-orthogonal matrices of a given order and expand the possibilities of their practical application in communication systems, coding, and digital information masking.

Keywords: orthogonal matrices, quasi-orthogonal matrices, Hadamard matrices, Mersenne matrices, matrix transformations.