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Non-split Toric Codes

  • Coding Theory
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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.

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References

  1. Advances in Algebraic Geometry Codes, Martínez-Moro, E., Munuera, C., and Ruano, D., Eds., Singapore: World Sci., 2008.

    MATH  Google Scholar 

  2. Cox, D.A., Little, J.B., and Schenck, H.K., Toric Varieties, Providence, R.A.: Amer. Math. Soc., 2011.

    Book  MATH  Google Scholar 

  3. Hansen, J.P., Toric Surfaces and Error-Correcting Codes, Coding Theory, Cryptography and Related Areas (Proc. Int. Conf. on Coding Theory, Cryptography and Related Areas, held in Guanajuato, Mexico, in April 1998), Buchmann, J., Høholdt, T., Stichtenoth, H., and Tapia-Recillas, H., Eds., Berlin: Springer, 2000, pp. 132–142.

    Chapter  Google Scholar 

  4. Hansen, J.P., Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes, Appl. Algebra Engrg. Comm. Comput., 2002, vol. 13, no. 4, pp. 289–300.

    Article  MathSciNet  MATH  Google Scholar 

  5. Berman, S.D., On the Theory of Group Codes, Kibernetika (Kiev), 1967, vol. 3, no. 1, pp. 31–39 [Cybernetics (Engl. Transl.), 1967, vol. 3, no. 1, pp. 25–31].

    MathSciNet  MATH  Google Scholar 

  6. Berman, S.D., Semisimple Cyclic and Abelian Codes. II, Kibernetika (Kiev), 1967, vol. 3, no. 3, pp. 21–30 [Cybernetics (Engl. Transl.), 1967, vol. 3, no. 3, pp. 17–23].

    MathSciNet  MATH  Google Scholar 

  7. Joyner, D., Toric Codes over Finite Fields, Appl. Algebra Engrg. Comm. Comput., 2004, vol. 15, no. 1, pp. 63–79.

    Article  MathSciNet  MATH  Google Scholar 

  8. Voskresenskii, V.E., Algebraic Groups and Their Birational Invariants, Providence, R.A.: Amer. Math. Soc., 1998.

    MATH  Google Scholar 

  9. Tsfasman, M.A., Vlăduţ, S.G., and Nogin, D.Yu., Algebrogeometricheskie kody. Osnovnye ponyatiya, Moscow: MCCME, 2003. Translated under the title Algebraic Geometric Codes: Basic Notions, Providence, R.I.: Amer. Math. Soc., 2007.

    Google Scholar 

  10. Massey, J.L., Reversible Codes, Inform. Control, 1964, vol. 7, no. 3, pp. 369–380.

    Article  MathSciNet  MATH  Google Scholar 

  11. Lachaud, G., The Parameters of Projective Reed-Müller Codes, Discrete Math., 1990, vol. 81, no. 2, pp. 217–221.

    Article  MathSciNet  MATH  Google Scholar 

  12. Voskresenskii, V.E., Projective Invariant Demazure Models, Izv. Akad. Nauk SSSR Ser. Mat., 1982, vol. 46, no. 2, pp. 195–210 [Math. USSR Izv. (Engl. Transl.), 1983, vol. 20, no. 2, pp. 189–202].

    MathSciNet  Google Scholar 

  13. Grassl, M., Bounds on the Minimum Distance of Linear Codes and Quantum Codes (electronic tables). Available at http://www.codetables.de (accessed on October 12, 2018).

  14. Platonov, V.P. and Rapinchuk, A.S., Algebraicheskie gruppy i teoriya chisel, Moscow: Nauka, 1991. Translated under the title Algebraic Groups and Number Theory, Boston: Academic, 1994.

    Google Scholar 

  15. Rubin, K. and Silverberg, A., Compression in Finite Fields and Torus-Based Cryptography, SIAM J. Comput., 2008, vol. 37, no. 5, pp. 1401–1428.

    Article  MathSciNet  MATH  Google Scholar 

  16. Voskresenskii, V.E., On Two-Dimensional Algebraic Tori, Izv. Akad. Nauk SSSR Ser. Mat., 1965, vol. 29, no. 1, pp. 239–244.

    MathSciNet  Google Scholar 

  17. Voskresenskii, V.E., On Two-Dimensional Algebraic Tori. II, Izv. Akad. Nauk SSSR Ser. Mat., 1967, vol. 31, no. 3, pp. 711–716 [Math. USSR Izv. (Engl. Transl.), 1967, vol. 1, no. 3, pp. 691–696].

    MathSciNet  MATH  Google Scholar 

  18. Graber, T., Harris, J., Mazur, B., and Starr, J., Arithmetic Questions Related to Rationally Connected Varieties, The Legacy of Niels Henrik Abel (The Abel Bicentennial Conf., Oslo, Norway, June 3–8, 2002), Laudal, O.A., Ed., Berlin: Springer, 2004, pp. 531–542.

    Chapter  Google Scholar 

  19. Voskresenskii, V.E. and Klyachko, A.A., Toroidal Fano Varieties and Root Systems, Izv. Akad. Nauk SSSR Ser. Mat., 1984, vol. 48, no. 2, pp. 237–263 [Math. USSR Izv. (Engl. Transl.), 1985, vol. 24, no. 2, pp. 221–244].

    MathSciNet  Google Scholar 

  20. Batyrev, V.V. and Tschinkel, Y., Rational Points of Bounded Height on Compactifications of Anisotropic Tori, Int. Math. Res. Notices, 1995, vol. 1995, no. 2, pp. 591–635.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ballard, M.R., Duncan, A., and McFaddin, P.K., On Derived Categories of Arithmetic Toric Varieties, arXiv:1709.03574v3 [math.AG], 2018.

  22. Poonen, B., Rational Points on Varieties, Providence, R.A.: Amer. Math. Soc., 2017.

    Book  MATH  Google Scholar 

  23. Hirschfeld, J.W.P., Finite Projective Spaces of Three Dimensions, Oxford: Oxford Univ. Press, 1986.

    Google Scholar 

  24. Couvreur, A., Construction of Rational Surfaces Yielding Good Codes, Finite Fields Appl., 2011, vol. 17, no. 5, pp. 424–441.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kollár, J., Looking for Rational Curves on Cubic Hypersurfaces, Higher-Dimensional Geometry over Finite Fields (Proc. of the NATO Advanced Study Institute, Göttingen, Germany, June 25–July 6, 2007), Kaledin, D. and Tschinkel Y., Eds., Amsterdam: IOS Press, 2008, pp. 92–122.

    Google Scholar 

  26. Hartshorne, R., Algebraic Geometry, Berlin: Springer, 1977. Translated under the title Algebraicheskaya geometriya, Moscow: Mir, 1981.

    Book  MATH  Google Scholar 

  27. Hernando, F., O’Sullivan, M.E., Popovici, E., and Srivastava, S., Subfield-Subcodes of Generalized Toric Codes, in Proc. 2010 IEEE Int. Sympos. on Information Theory (ISIT’2010), Austin, TX, USA, June 13–18, 2010, pp. 1125–1129.

  28. Massey, J.L., Linear Codes with Complementary Duals, Discrete Math., 1992, vol. 106–107, pp. 337–342.

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, X. and Massey, J.L., The Condition for a Cyclic Code to Have a Complementary Dual, Discrete Math., 1994, vol. 126, no. 1–3, pp. 391–393.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kasami, T., Lin, S., and Peterson, W., New Generalizations of the Reed-Muller Codes. I: Primitive Codes, IEEE Trans. Inform. Theory, 1968, vol. 14, no. 2, pp. 189–199.

    Article  MathSciNet  MATH  Google Scholar 

  31. Couvreur, A. and Duursma, I., Evaluation Codes from Smooth Quadric Surfaces and Twisted Segre Varieties, Des. Codes Cryptogr., 2013, vol. 66, no. 1–3, pp. 291–303.

    Article  MathSciNet  MATH  Google Scholar 

  32. Boguslavsky, M.I., Sections of the Del Pezzo Surfaces and Generalized Weights, Probl. Peredachi Inf., 1998, vol. 34, no. 1, pp. 18–29 [Probl. Inf. Transm. (Engl. Transl.), 1998, vol. 34, no. 1, pp. 14–24].

    MathSciNet  Google Scholar 

  33. Ruano, D., On the Parameters of r-Dimensional Toric Codes, Finite Fields Appl., 2007, vol. 13, no. 4, pp. 962–976.

    Article  MathSciNet  MATH  Google Scholar 

  34. Aubry, Y. and Perret, M., A Weil Theorem for Singular Curves, Arithmetic, Geometry and Coding Theory (Proc. Int. Conf. held at Luminy, France, June 28–July 2, 1993), Pellikaan, R., Perret, M., and Vlăduţ, S., Eds., Berlin: de Gruyter, 1996, pp. 1–7.

    Google Scholar 

  35. Daskalov, R. and Hristov, P., New One-Generator Quasi-cyclic Codes over GF(7), Probl. Peredachi Inf., 2002, vol. 38, no. 1, pp. 59–63 [Probl. Inf. Transm. (Engl. Transl.), 2002, vol. 38, no. 1, pp. 50–54].

    MathSciNet  MATH  Google Scholar 

  36. Daskalov, R. and Hristov, P., New Quasi-cyclic Degenerate Linear Codes over GF(8), Probl. Peredachi Inf., 2003, vol. 39, no. 2, pp. 29–35 [Probl. Inf. Transm. (Engl. Transl.), 2003, vol. 39, no. 2, pp. 184–190].

    MathSciNet  Google Scholar 

  37. Daskalov, R., Metodieva, E., and Hristov, P., New Minimum Distance Bounds for Linear Codes over GF(9), Probl. Peredachi Inf., 2004, vol. 40, no. 1, pp. 15–26 [Probl. Inf. Transm. (Engl. Transl.), 2004, vol. 40, no. 1, pp. 13–24].

    MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Algebra and Number Theory Laboratory, Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    D. I. Koshelev

  2. Department of Discrete Mathematics, Moscow Institute of Physics and Technology (State University), Moscow, Russia

    D. I. Koshelev

  3. Versailles Laboratory of Mathematics, Versailles Saint-Quentin-en-Yvelines University, Versailles, France

    D. I. Koshelev

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Correspondence to D. I. Koshelev.

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Russian Text © The Author(s), 2019, published in Problemy Peredachi Informatsii, 2019, Vol. 55, No. 2, pp. 28–49.

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Koshelev, D.I. Non-split Toric Codes. Probl Inf Transm 55, 124–144 (2019). https://doi.org/10.1134/S0032946019020029

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