Syndrome spectrums of error orbits in RS-codes
https://doi.org/10.21122/2309-4923-2020-1-34-38
Abstract
This article is devoted to the research of the properties of syndromes of errors in Reed-Solomon codes. RS-codes are built on non-binary alphabets. So, unlike BCH-codes, RS-codes contain an extremely large variety of correctable errors. To correct these errors, a systematic application of automorphisms of codes is proposed. Characteristic automorphisms of RS-codes are cyclic and affine substitutions forming cyclic groups Г and A whose orders coincide with the code length. Cyclic and affine substitutions commute with each other and generate a joint АГ group, what is the product of subgroups A and Г. These three groups act on the space of error vectors of RS-codes, breaking this space into three types of error orbits. As a rule, these orbits are complete and contain the maximum possible number of errors. Syndromes are the main indicator of the presence of errors in each message received by the information system, a means of accurately identifying these errors. The specificity of syndromes of double errors in RS-codes is investigated. Determined that syndrome spectrums of error orbits are also complete in most cases. Proved that the structure of the syndrome spectrums copies the structure of the orbits themselves, which in turn copy the structure of groups of code automorphisms. The results obtained are a significant contribution to the construction of the theory of syndrome norms for RS-codes.
About the Authors
V. A. Lipnitski
Military Academy of the Republic of Belarus
Belarus
Belarus
Lipnitski V. А., Doctor of technical sciences, professor
Minsk
S. I. Semyonov
Military Academy of the Republic of Belarus
Belarus
Belarus
Semyonov S. I., the adjunct of chair of information and computing systems
Minsk
References
1. McWilliams F. J., Sloan J. J. The Theory of ErrorCorrecting Codes. – Amsterdam: northholland publishing compa- ny, 1977. – 762 s.
2. Sclyar B. Digital communications. Fundamentals and Applications. Ed. 2. – Moscow: Wil’ams; 2003–1104 s.
3. Kudryashov B. D. Fundamentals of coding theory. – St. Petersburg: BHVPetersburg; 2016–400 s.
4. Marov A. V., Uteshev A. Y. Matrix formalism of PeedSolomon codes // Vesnik SanctPeterburgskogo universiteta, Ser.10, V.4. – St. Petersburg: BHVPetersburg; 2016 – s. 4–17.
5. Lipnitsky V. А. Konopelko V. К. Norm decoding of noiseresistant codes and algebraic equations. – Minsk: BGU; 2007–239 s.
6. Lipnitsky V. А., Semyonov S. I. The automorphisms and error orbits of ReedSolomon codes // Doklady BGUIR V. 6 – Minsk: BGUIR, 2019. (In Russ.)
Review
For citations:
Lipnitski V.A., Semyonov S.I. Syndrome spectrums of error orbits in RS-codes. «System analysis and applied information science». 2020;(1):34-38. (In Russ.) https://doi.org/10.21122/2309-4923-2020-1-34-38
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