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
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 193)


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.


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



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


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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

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