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Telecommunications and Radio Engineering
SJR: 0.202 SNIP: 0.2 CiteScore™: 0.23

ISSN Imprimer: 0040-2508
ISSN En ligne: 1943-6009

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Telecommunications and Radio Engineering

DOI: 10.1615/TelecomRadEng.v78.i5.30
pages 393-417

ANALYSIS AND STUDYING OF THE PROPERTIES OF ALGEBRAIC GEOMETRIC CODES

A. A. Kuznetsov
V. Karazin National University of Kharkiv, 4 Svobody Sq., Kharkiv 61022, Ukraine
I. P. Kolovanova
V. Karazin National University of Kharkiv, 4 Svobody Sq., Kharkiv 61022, Ukraine
D. I. Prokopovych-Tkachenko
University of Customs and Finance, 2/4 Volodymyra Vernadskoho St., Dnipro 49000, Ukraine
T. Y. Kuznetsova
Institute of Information Technologies, 12 Bakulina St., Kharkiv 61166, Ukraine

RÉSUMÉ

Linear block noise-proof codes constructed according to algebraic curves (algebraic geometric codes) are considered, their design properties are evaluated, algorithms of construction and decoding are studied. The energy efficiency of the transmission of discrete messages by M-ary orthogonal signals in the application of algebraic geometric codes is studied; the achievable energy gain from the use of noise-immune coding is estimated. It is shown that in discrete channels without memory it is possible to obtain a significant energy gain, which increases with the transition to long algebraic geometric codes constructed by curves with a large number of points with respect to the genus of the curve. It is established that the computational complexity of implementing algebraic geometric codes is comparable to other known noise-resistant codes, for example, the Reed-Solomon codes and others. Thus, high energy efficiency in combination with acceptable computational complexity of implementation confirm the prospects of algebraic geometric codes using in modern telecommunication systems and networks to improve the noise immunity of data transmission channels.

RÉFÉRENCES

  1. Goppa, V.D., (1981) , Codes on algebraic curves, Dokl. Akad. Nauk SSSR, 259(6), pp. 1289-1290, (in Russian).

  2. Goppa, V.D., (1984) , Codes and information, Achievements of Mathematical Sciences, 30(235), pp. 77-120, (in Russian).

  3. Tsfasman, М.А., (1982) , Goppa codes that are better than the Varshamov-Gilbert bound, Problems of Information Transmission, 18(3), pp. 3-6, (in Russian).

  4. Shafarevich, I.R., (1972) , Basic Algebraic Geometry, Moscow, Russia: Nauka, 568 p., (in Russian).

  5. Stein, S. and Jones, J., (1971) , Modern Communication Principles: with Application to Digital Signaling, Moscow, Russia: Svyaz, 376 p., (in Russian).

  6. Kasami, Т., Тokura, N., Ivadari, Е., and Inagaki, J., (1978) , Coding Theory, Moscow, Russia: Mir, 576 p., (in Russian).

  7. Blahut, R., (1986) , Theory and Practice of Error Control Codes, Moscow, Russia: Mir, 576 p., (in Russian).

  8. MacWilliams, F.J. and Sloane, N.J.А., (1979) , Theory of Error-Correcting Codes, Moscow, Russia: Svyaz, 744 p., (in Russian).

  9. Feng, G.L. and Rao, T.R.N., (1993) , Decoding algebraic geometric codes up to the designed minimum distance, IEEE Trans. Inform. Theory, 39(1), pp. 37-46.

  10. Sakata, S., Justesen, J., Madelung, Y., Jensen, H.E., and Hoholdt, T., (1995) , Fast Decoding of Algebraic-Geometric Codes up to the Designed Minimum Distance, IEEE Trans. Inform. Theory, 41(5), pp. 1672-1677.

  11. Olshevksy, V. and Shokrollahi, A., (1999) , A displacement structure approach to decoding algebraic geometric codes, Proceedings of the 31st annual ACM Symposium on Theory of Computing (STOC), pp. 235-244.


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