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

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Intelligent Decision Technologies (IDT 2020)

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

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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.

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Acknowledgement

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

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Authors and Affiliations

  1. Saint-Petersburg State University of Aerospace Instrumentation, 67 Bolshaya Morskaya Street, Saint-Petersburg, 190000, Russian Federation

    Alexander Sergeev, Mikhail Sergeev, Nikolaj Balonin & Anton Vostrikov

Authors
  1. Alexander Sergeev
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  2. Mikhail Sergeev
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  3. Nikolaj Balonin
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  4. Anton Vostrikov
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Corresponding author

Correspondence to Anton Vostrikov .

Editor information

Editors and Affiliations

  1. Gdynia Maritime University, Gdynia, Poland

    Ireneusz Czarnowski

  2. KES International Research, UK

    Robert J. Howlett

  3. Faculty of Engineering and Information Technology, Centre for Artificial Intelligence, University of Technology Sydney, Sydney, NSW, Australia

    Lakhmi C. Jain

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

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Sergeev, A., Sergeev, M., Balonin, N., Vostrikov, A. (2020). Symmetry Indices as a Key to Finding Matrices of Cyclic Structure for Noise-Immune Coding. In: Czarnowski, I., Howlett, R., Jain, L. (eds) Intelligent Decision Technologies. IDT 2020. Smart Innovation, Systems and Technologies, vol 193. Springer, Singapore. https://doi.org/10.1007/978-981-15-5925-9_19

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