On Relationships among Quasi-orthogonal Matrices Constructed on the Known Sequences of Prime Numbers

Abstract

The objective of the paper is to show the relationship among numbers, belonging to known sequences of prime numbers, and quasi-orthogonal matrices, existing for orders equal to these numbers, and the relationship among such matrices based on calculating algorithms. Methods: analysis of the sequences of quasi-orthogonal matrices with absolute and local maximum of its determinant, detecting of the structural invariants in the matrices, matching algorithms for calculating these matrices. Results: known sequences of natural numbers are considered, the definition of a matrix, associated with a natural number, is formulated. Sequences of numbers with proved existence of quasi-orthogonal matrices associated with them are presented. It is suggested that associated matrices exist for all positive natural numbers. Properties of these types of matrices, their relationships based on calculating algorithms are considered. There are modified algorithms and general key chains of Euler and Mersenne matrices presented, a sequence of orders of which are systemically important. Practical value: quasi-orthogonal matrices of absolute and local maximum of determinant have immediate practical value for error-correcting coding tasks, video compression and masking. Their diversity allows developers of technical systems greatly facilitate a matrix selection, optimal one for a particular task.