MERSENNE AND HADAMARD MATRICES CALCULATION BY SCARPIS METHOD

N. A. Balonin, Y. N. Balonin, M. B. Sergeev


Read the full article 
Article in Russian


Abstract

Purpose. The paper deals with the problem of basic generalizations of Hadamard matrices associated with maximum determinant matrices or not optimal by determinant matrices with orthogonal columns (weighing matrices, Mersenne and Euler matrices, ets.); calculation methods for the quasi-orthogonal local maximum determinant Mersenne matrices are not studied enough sufficiently. The goal of this paper is to develop the theory of Mersenne and Hadamard matrices on the base of generalized Scarpis method research.
Methods. Extreme solutions are found in general by minimization of maximum for absolute values of the elements of studied matrices followed by their subsequent classification according to the quantity of levels and their values depending on orders. Less universal but more effective methods are based on structural invariants of quasi-orthogonal matrices (Silvester, Paley, Scarpis methods, ets.).
Results. Generalizations of Hadamard and Belevitch matrices as a family of quasi-orthogonal matrices of odd orders are observed; they include, in particular, two-level Mersenne matrices. Definitions of section and layer on the set of generalized matrices are proposed. Calculation algorithms for matrices of adjacent layers and sections by matrices of lower orders are described. Approximation examples of the Belevitch matrix structures up to 22-nd critical order by Mersenne matrix of the third order are given. New formulation of the modified Scarpis method to approximate Hadamard matrices of high orders by lower order Mersenne matrices is proposed. Williamson method is described by example of one modular level matrices approximation by matrices with a small number of levels.
Practical relevance. The efficiency of developing direction for the band-pass filters creation is justified. Algorithms for Mersenne matrices design by Scarpis method are used in developing software of the research program complex. Mersenne filters are based on the suboptimal by determinant matrices and are used for image masking and compression.


Keywords: orthogonal matrices, Hadamard matrices, Belevitch matrices, Mersenne matrices, Mersenne numbers, Scarpis method, Williamson array, video information protection.

References
 1.     Mironovsky L.A., Slaev V.A. Strip-metod preobrazovaniya izobrazhenii i signalov [The strip-method for transforming signals and images]. St. Petersburg, Politekhnika Publ., 2006, 163 p.
2.     Erosh I.L., Sergeev A.M., Filatov G.P. O zashchite tsifrovykh izobrazhenii pri peredache po kanalam svyazi [Protection of images during transfer via communication channels]. Informatsionno-Upravlyayushchie Sistemy, 2007, no. 5,
pp. 20–22.
3.     Balonin Yu.N., Vostrikov A.A., Sergeev M.B. O prikladnykh aspektakh primeneniya M-matrits [Applied aspects of M-matrix use]. Informatsionno-Upravlyayushchie Sistemy, 2012, no. 1, pp. 92–93.
4.     Balonin Yu.N., Sergeev M.B. Algoritm i programma poiska i issledovaniya M-matrits [The algorithm and program of M-matrices search and study]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2013, no. 3 (85), pp. 82–86.
5.     Balonin N.A., Sergeev M.B. M-matritsy [М-Matrices]. Informatsionno-Upravlyayushchie sistemy, 2011, no. 1,
pp. 14–21.
6.     Hadamard J. Résolution d'une question relative aux determinants. Bulletin des Sciences Mathématiques, 1893, vol. 17, pp. 240–246.
7.     Balonin N.A., Sergeev M.B., Mironovsky L.A. Vychislenie matrits Adamara-Mersenna [Calculation of Hadamard-Mersenne matrices]. Informatsionno-Upravlyayushchie Sistemy, 2012, no. 5, pp. 92–94.
8.     Balonin N.A., Sergeev M.B. O dvukh sposobakh postroeniya matrits Adamara-Eilera [Two ways to construct Hadamard-Euler matrices]. Informatsionno-Upravlyayushchie Sistemy, 2013, no. 1 (62), pp. 7–10.
9.     Balonin N.A., Sergeev M.B., Mironovsky L.A. Vychislenie matrits Adamara-Ferma [Calculation of Hadamard-Fermat matrices]. Informatsionno-Upravlyayushchie Sistemy, 2012, no. 6 (61), pp. 90–93.
10.  Scarpis U. Sui determinanti di valore Massimo. Rendiconti della R. Istituto Lombardo di Scienze e Lettere, 1898, vol. 31, pp. 1441–1446.
11.  Paley R.E.A.C. On orthogonal matrices. Journal of Mathematics and Physics, 1933, vol. 12, pp. 311–320.
12.  Belevitch V. Theorem of 2n-terminal networks with application to conference telephony. Electronic Communications,1950, vol. 26, pp. 231–244.
13.  Balonin Yu.N., Sergeev M.B. М-matritsa 22-go poryadka [M-matrix of the 22nd order]. Informatsionno-Upravlyayushchie Sistemy, 2011, no. 5, pp. 87–90.
14.  Williamson J. Hadamard’s determinant theorem and the sum of four squares. Duke Math. J, 1944, vol. 11, pp. 65–81.
15.  Balonin N.A., Sergeev M.B. O rasshirenii ortogonal'nogo bazisa v zadachakh szhatiya videoizobrazhenii [Expansion of the orthogonal basis in video compression]. Vestnik Komp’iuternykh i Informatsionnykh Tekhnologii, 2014, no. 2 (116), pp. 11–15.
16.  Vostrikov A.A., Balonin Yu.N. Matritsy Adamara-Mersenna kak bazis ortogonal'nylh preobrazovanii v maskirovanii videoizobrazhenii [Hadamard – Mersenne matrices as a basis of orthogonal transformation for video masking encoding]. Izv. vuzov. Priborostroenie, 2014, vol. 57, no. 1, pp. 15–19.
17.  Balonin N.A., Sergeev M.B. Matritsy local'nogo maksimuma determinanta [Local maximum determinant matrices]. Informatsionno-Upravlyayushchie Sistemy, 2014, no. 1 (68), pp. 2–15.
18.  Balonin N.A., Sergeev M.B. Matritsa zolotogo secheniya G10 [Matrix of golden ratio G10]. Informatsionno-Upravlyayushchie Sistemy, 2013, no. 6 (67), pp. 2–5.
19.  Balonin N.A., Sergeev M.B. M-matritsy i kristallicheskie struktury [M-matrices and crystal structures]. Vestnik of Nosov Magnitogorsk State Technical University, 2013, no. 3 (43), pp. 58–62.
Copyright 2001-2017 ©
Scientific and Technical Journal
of Information Technologies, Mechanics and Optics.
All rights reserved.

Яндекс.Метрика