Dynamic Generators of the Quasiortogonal Hadamard Matrix Family
Abstract
The problem of constructing non-linear and linear finite-field generators of quasi-orthogonal matrices of the Hadamard family with a small number of distinct values of their elements not exceeding by absolute value 1 and a global or local maximum of determinant is investigated. The properties of such dynamical systems are analyzed; the classification of the matrix families and their ornaments, obtained with their help, is described; the way of proving the existence of real and integer matrices different from the combinatorial approach is shown. The values to which the elements of the matrix are equal are called its levels. The concepts of the Hadamard norm and the determinant of a quasiorthogonal matrix are introduced. Levels, Hadamard norm and determinant play a fundamental role in the definitions of classes of generalized matrices of the Hadamard family. The classes of the Hadamard, Belevich (conference matrices), Seberri (weighing matrices), Mersenne, Euler, Odin (Seidel), Fermat matrices are described. The formulas for the values of their levels are given. Ornaments of Euler matrices answer to the question of the maximum complexity of Hadamard matrices — two border two circulant structure.References
Published
2017-10-12
How to Cite
Balonin, N., Sergeev, M., & Suzdal, V. (2017). Dynamic Generators of the Quasiortogonal Hadamard Matrix Family. SPIIRAS Proceedings, 5(54), 224-243. https://doi.org/10.15622/sp.54.10
Section
Theoretical and Applied Mathematics
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
