Two-Circulant M-Matrix of the 22nd Order
Keywords:
Orthogonal Matrices, Hadamard Matrices, Conference Matrices, Belevitch Matrices, Weighing Matrices, Matrix Determinant, Two-Circulant Matrices, Golay Codes, Barker CodesAbstract
Purpose: Within a class of quasi-orthogonal matrices of even orders different from Hadamard matrices a nature of the optimal by
determinant solutions depends on an amount of zeros in columns of weighing matrices. Two-circulant forms of optimal or suboptimal
matrices have been used as a source of complementary pair sequences generalizing Golay and Barker codes. The goal of this paper is to
construct an example of a suboptimal two-circulant matrix of the 22nd order which is critical for conference matrices. Methods: There
have been found extreme solutions by minimization of maximum of matrix elements absolute values with their consequent classification
according to an amount and values of levels depending on orders. Results: There has been identified and described by matrix portrait and
meaning of levels a modular two-circulant six-level quasi-orthogonal matrix of the 22nd order (M-matrix) having the local maximum.
There has been formulated a conjecture on replacing Belevitch matrices if they are absent (do not exist due to the known criteria) by
odd order M-matrices. To demonstrate the abovementioned there has been given a comparison of two-circulant 22nd order M-matrix
and a weighing matrix W(22,20) by its structures and determinants. Practical relevance: The algorithms of constructing two-circulant
M-matrices have been applied for development of the research software. Generalized complementary pair sequences have been used as a
basis of filers for image masking and compression.