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Problems of Information Transmission

, Volume 55, Issue 2, pp 124–144 | Cite as

Non-split Toric Codes

  • D. I. KoshelevEmail author
Coding Theory

Abstract

We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed-Solomon and (projective) Reed-Muller codes, are contained (up to equivalence) in the new class. Our codes are explicitly described in terms of algebraic and toric geometries over finite fields; therefore, they can easily be constructed in practice. Finally, we obtain new cyclic reversible codes, namely non-split toric codes on the del Pezzo surface of degree 6 and Picard number 1. We also compute their parameters, which prove to attain current lower bounds at least for small finite fields.

Key words

finite fields toric and cyclic codes non-split algebraic tori toric varieties del Pezzo surfaces elliptic curves 

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Notes

Acknowledgement

The author is deeply grateful to his scientific advisor M.A. Tsfasman and also to V. Batyrev, S. Gorchinskiy, G. Kabatiansky, B. Kunyavskii, K. Loginov, A. Perepechko, S. Rybakov, K. Shramov, V. Stukopin, D. Timashev, A. Trepalin, S. Vlİduţ, I. Vorobyev, and participants of the Coding Theory seminar run by L.A. Bassalygo at the Institute for Information Transmission Problems of the Russian Academy of Sciences for their help and useful comments.

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Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Algebra and Number Theory Laboratory, Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia
  2. 2.Department of Discrete MathematicsMoscow Institute of Physics and Technology (State University)MoscowRussia
  3. 3.Versailles Laboratory of MathematicsVersailles Saint-Quentin-en-Yvelines UniversityVersaillesFrance

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