Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers
https://doi.org/10.31854/1813-324X-2022-8-4-14-19
Abstract
Quasi-orthogonal Hadamard matrices and Mersenne matrices with two and three values of the elements, used in digital data processing, are considered, as well as the basis of error-correcting codes and algorithms for transforming orthogonal images. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The connection between symmetry and antisymmetry of structures of cyclic Hadamard and Mersenne matrices on a orders equal to prime numbers, products of close primes, composite numbers, powers of a prime number is shown. Separately, orders equal to the degrees of the prime number 2 are distinguished, both the orders of Hadamard matrices and the basis of the composite orders of Mersenne matrices of block structures with two element values. It is shown that symmetric Hadamard matrices of cyclic and bicyclic structures, according to the extended Riser boundary, do not exist on orders above 32. Mersenne matrices of composite orders belonging to the sequence of Mersenne numbers 2k ‒ 1 nested in the sequence of orders of the main family of Mersenne matrices 4t ‒ 1 exist in a symmetric and antisymmetric form. For orders equal to the powers of a prime number, Mersenne matrices exist in the form of block-diagonal constructions with three element values. The value of prime power determines the number of blocks along the diagonal of the matrix on which the elements with the third value are located. The cyclic blocks are symmetrical and antisymmetric.
Keywords
prime numbers,
quasi-orthogonal matrices,
Hadamard matrices,
Mersenne matrices,
symmetric and antisymmetric matrices,
signal encoding,
image encoding
Supporting agencies: This research was funded by RFBR according to the research project No. 19-29-06029.
About the Author
A. Sergeev
Saint-Petersburg State University of Aerospace Instrumentation
Russian Federation
Russian Federation
Aleksandr Sergeev
St. Petersburg, 190000
References
1. Seberry J., Yamada M. Hadamard Matrices: Constructions using Number Theory and Linear Algebra. Hoboken: John Wiley & Sons; 2020. 352 p.
2. Balonin N.A., Sergeev M.B. The generalized Hadamard matrix norms. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2014;2:5–11. (in Russ.)
3. Mohan M.T. p-almost Hadamard matrices and λ-planes. Journal of Algebraic Combinatorics. 2022;55(1):89‒108. DOI:10.1007/s10801-020-00991-y
4. Mironovsky L.A., Slaev V.A. Strip-Method for Image and Signal Transformation. Berlin, Boston: De Gruyter; 2011. 166 p. DOI:10.1515/9783110252569
5. Seberry J., Wysocki В.J., Wysocki Т.A. On some applications of Hadamard matrices. Metrika. 2005;62(2-3):221–239. DOI:10.1007/s00184-005-0415-y
6. Wang R. Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis. Cambridge: Cambridge University Press.; 2010. 504 p.
7. Balonin N., Vostrikov A., Sergeev M. Mersenne-Walsh Matrices for Image Processing. Proceedings of the 8th International KES Conference on Intelligent Interactive Multimedia: Systems and Services, KES-IIMSS-15, 17‒19 June 2015, Sorrento, Italy. Smart Innovation, Systems and Technologies. vol.40. Cham: Springer; 2015. p.141–147. DOI: 10.1007/978-3-319-19830-9_13
8. Sergeev M.B., Nenashev V.А., Sergeev А.М. Nested code sequences of Barker ‒ Mersenne ‒ Raghavarao. Informatsionnoupravliaiushchie sistemy. 2019;3(100):71–81. (in Russ.) DOI:10.31799/1684-8853-2019-3-71-81
9. Balonin N.A., Sergeev M.B. Golden ratio Matrix G10. Informatsionno-upravliaiushchie sistemy. 2013;6(67):2–5. (in Russ.)
10. Balonin N.A., Vostrikov A.A., Sergeev M.B. On two predictors of calculable chains of quasi-orthogonal matrices. Automatic Control and Computer Sciences. 2015;49(3):153–158. DOI:10.3103/S0146411615030025
11. Sergeev A.M. The interconnection of one kind of quasi-orthogonal matrices built on the orders of sequences 4k and 4k ‒ 1. Proceedings of Saint Petersburg Electrotechnical University. 2017;7:12–17. (in Russ.)
12. Sergeev A., Sergeev M., Balonin N., Vostrikov A. Symmetry Indices as a Key to Finding Matrices of Cyclic Structure for Noise-Immune Coding. Proceedings of the 12th KES International Conference on Intelligent Decision Technologies, KES-IDT 2020, 17‒19 June 2020). Smart Innovation, Systems and Technologies. vol.193. Singapore: Springer; 2020. p. 223–230. DOI:10.1007/978-981-15-5925-9_19
13. Ryser H.J. Combinatorial Mathematics. New York: John Wiley; 1963. 162 p.
14. Balonin N.A., Sergeev M.B. Ryser's Conjecture Expansion for Bicirculant Strictures and Hadamard Matrix Resolvability by Double-Border Bicycle Ornament. Informatsionno-upravliaiushchie sistemy. 2017;1(86):2–10. (in Russ.) DOI:10.15217/issnl684-8853.2017.1.2
15. Balonin N.A., Sergeev M.B. To the question of the existence of Mersenne and Hadamard matrices. Informatsionno-upravliaiushchie sistemy. 2013;5(66):2–8. (in Russ.)
16. Sergeev A., Sergeev M., Vostrikov A., Kurtyanik D. Portraits of Orthogonal Matrices as a Base for Discrete Textile Ornament Patterns. Smart Innovation, Systems and Technologies. 2019. Vol. 143. P. 135 – 143. DOI:10.1007/978-981-13-8303-8_12
17. Zenkin V.I. Prime number distribution: Elementary methods. Kaliningrad; 2008. (in Russ.) URL: http://regiomontan.ru/book/VZ_primes.pdf [Accessed 14th August 2022]
18. Hall M., Jr. A Survey of Difference Sets. Proceedings of the American Mathematical Society. 1956;7(6):975–986. DOI:10.2307/2033024
19. Mathscinet. One circulant Mersenne Matrices. URL: http://mathscinet.ru/catalogue/CM1 [Accessed 14th August 2022]
20. Vostrikov A., Sergeev A., Balonin Y. Using Families of Extremal Quasi-Orthogonal Matrices in Communication Systems. Proceedings of the 13th KES Conference on Intelligent Decision Technologies, KES-IDT 2021, 14‒16 June 2020. Smart Innovation, Systems and Technologies. vol.238. Singapore: Springer; 2021. p.95–108. DOI:10.1007/978-981-16-2765-1_8
21. Vostrikov А., Chernyshev S. On distortion assessment of images masking with M-matrices. Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 2013;5(87):99–103. (in Russ.)
22. Mironovsky L.A., Slaev V.A. The strip method of noise-immune image transformation. Measurement Techniques. 2006;49:745–754. (in Russ.) DOI:10.1007/s11018-006-0183-8
23. Balonin Y., Vostrikov A., Sergeev M. Software for finding M-matrices. Frontiers in Artificial Intelligence and Applications. 2014;262:475–480. DOI:10.3233/978-1-61499-405-3-475
24. Grigoriev E. Study of Correlation Properties of New Code Sequences Based on Persymmetric Quasi-Orthogonal Circulants. Proc. of Telecom. Universities. 2022;8(2):83‒90. (in Russ.) DOI:10.31854/1813-324X-2022-8-2-83-90
25. Roshan P., Leary J. 802.11 Wireless LAN Fundamentals. Cisco Press; 2004. 312 p.
Review
For citations:
Sergeev A. Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers. Proceedings of Telecommunication Universities. 2022;8(4):14-19. (In Russ.) https://doi.org/10.31854/1813-324X-2022-8-4-14-19
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