Advertisement

Symmetry Indices as a Key to Finding Matrices of Cyclic Structure for Noise-Immune Coding

  • Alexander Sergeev
  • Mikhail Sergeev
  • Nikolaj Balonin
  • Anton VostrikovEmail author
Conference paper
  • 43 Downloads
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 193)

Abstract

The paper discusses methods for assessing the symmetries of Hadamard matrices and special quasi-orthogonal matrices of circulant and two circulant structures used as the basis for searching for noise-resistant codes. Such codes, obtained from matrix rows intended for use in open communications, expand the basic and general theory of signal coding and ensure that the requirements for contemporary telecommunication systems are met. Definitions of the indices of symmetry, asymmetry, and symmetry defect of special matrices are given. The connection of symmetric and antisymmetric circulant matrices with primes, compound numbers, and powers of a prime number is shown. Examples of two circulant matrices that are optimal by their determinant, as well as special circulant matrices, are given. The maximum orders of the considered matrices of symmetric structures are determined.

Keywords

Orthogonal matrices Quasi-orthogonal matrices Hadamard matrices Raghavarao matrices Mersenne matrices Noise-immune coding Special symmetric matrices Antisymmetric matrices 

Notes

Acknowledgement

The reported study was funded by RFBR, project number 19-29-06029.

References

  1. 1.
    Wang,, R.: Introduction to Orthogonal Transforms with Applications in Data Processing and Analysis, p. 504. Cambridge University Press (2010)Google Scholar
  2. 2.
    Ahmed, N., Rao, K.R.: Orthogonal Transforms for Digital Signal Processing. p. 263. Springer Berlin (1975)Google Scholar
  3. 3.
    Klemm, R. (ed.): Novel Radar Techniques and Applications, vol. 1, Real Aperture Array Radar, Imaging Radar, and Passive and Multistatic Radar, p. 951. Scitech Publishing, London (2017)Google Scholar
  4. 4.
    NagaJyothi, A., Rajeswari, K.: Raja Generation and Implementation of Barker and Nested Binary Codes. J. Electr. Electron. Eng. 8(2), 33–41 (2013)Google Scholar
  5. 5.
    Proakis, J., Salehi M.: Digital communications, P. 1170. McGraw-Hill, Singapore (2008)Google Scholar
  6. 6.
    Levanon, N., Mozeson, E.: Radar Signals. p. 411. Wiley (2004)Google Scholar
  7. 7.
    IEEE802.11 Official site – URL: www.ieee802.org
  8. 8.
    Borwein, P., Mossinghoff, M.: Wieferich pairs and Barker sequences, II. LMS J. Comput. Math. 17(1), 24–32 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sergeev, A., Nenashev, V., Vostrikov, A., Shepeta, A., Kurtyanik, D.: Discovering and Analyzing Binary Codes Based on Monocyclic Quasi-Orthogonal Matrices. Smart Innov., Syst. Technol. 143, 113–123 (2019)CrossRefGoogle Scholar
  10. 10.
    Sergeev, M.B., Nenashev V.A., Sergeev A.M.: Nested code sequences of Barker-Mersenne-Raghavarao, Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 3, 71–81 (In Russian) (2019)Google Scholar
  11. 11.
    Ryser, H.J.: Combinatorial mathematics. The Carus Mathematical Monographs, no. 14. Published by The Mathematical Association of America; distributed by Wiley, New York (1963)Google Scholar
  12. 12.
    Schmidt, B.: Towards Ryser’s Conjecture. European Congress of Mathematics. Progress in Mathematics, vol 201, pp. 533–541. Birkhäuser, Basel (2001)Google Scholar
  13. 13.
    Euler, R., Gallardo, L.H., Rahavandrainy, O.: Eigenvalues of circulant matrices and aconjecture of ryser. Kragujevac J. Math.- 45(5), 751–759 (2021)Google Scholar
  14. 14.
    Balonin, N.A., Sergeev, M.B.: Quasi-orthogonal local maximum determinant matrices. Appl. Math. Sci. 9(6), 285–293 (2015)Google Scholar
  15. 15.
    Sergeev, A.M.: Generalized mersenne matrices and balonin’s conjecture. Autom. Control Comput. Sci. 48(4), 214–220 (2014)CrossRefGoogle Scholar
  16. 16.
    Balonin, N.A., Djokovic, D.Z.: Symmetry of Two-Circulant Hadamard Matrices and Periodic Golay Pairs, Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 3. 2–16 (2015)Google Scholar
  17. 17.
    Sergeev, A., Blaunstein N.: Orthogonal Matrices with Symmetrical Structures for Image Processing, Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 6(91), 2–8 (2017)Google Scholar
  18. 18.
    Hall, M.: A survey of difference sets. Proc. Amer. Math. Soc. 7, 975–986 (1956)Google Scholar
  19. 19.
    Hadamard, J.: Resolution d’une question relative aux determinants. Bull. des Sci. Math. 17, 240–246 (1893)zbMATHGoogle Scholar
  20. 20.
    Balonin, N.A., Sergeev M.B.: Weighted Conference Matrix Generalizing Belevich Matrix at the 22nd Order, Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 5, 97–98 (2013)Google Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Saint-Petersburg State University of Aerospace InstrumentationSaint-PetersburgRussian Federation

Personalised recommendations