Advertisement

Problems of Information Transmission

, Volume 46, Issue 4, pp 321–345 | Cite as

Special sequences as subcodes of reed-solomon codes

  • A. A. DavydovEmail author
  • V. V. Zyablov
  • R. E. Kalimullin
Coding Theory

Abstract

We consider sequences in which every symbol of an alphabet occurs at most once. We construct families of such sequences as nonlinear subcodes of a q-ary [n, k, n − k + 1] q Reed-Solomon code of length nq consisting of words that have no identical symbols. We introduce the notion of a bunch of words of a linear code. For dimensions k ≤ 3 we obtain constructive lower estimates (tight bounds in a number of cases) on the maximum cardinality of a subcode for various n and q, and construct subsets of words meeting these estimates and bounds. We define codes with words that have no identical symbols, observe their relation to permutation codes, and state an optimization problem for them.

Keywords

Generator Matrix Information Transmission Linear Code Generator Matrice Base Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Varakin, L.E., Sistemy svyazi s shumopodobnymi signalami (Communication Systems with Noise-like Signals),:, 1985.Google Scholar
  2. 2.
    Chu, W., Colbourn, C.J., and Dukes, P., Constructions for Permutation Codes in Powerline Communications, Des. Codes Cryptogr., 2004, vol. 32, no. 1–3, pp. 51–64.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dukes, P.J., Permutation Codes and Arrays, Section VI.44 of Handbook of Combinatorial Designs, Colbourn, C.J. and Dinitz, J.H., Eds., Boca Raton: Chapman & Hall, 2007, 2nd ed., pp. 568–571.Google Scholar
  4. 4.
    MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.zbMATHGoogle Scholar
  5. 5.
    Roth, R.M. and Seroussi, G., On Generator Matrices of MDS Codes, IEEE Trans. Inform. Theory, 1985, vol. 31, no. 6, pp. 826–830.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Roth, R.M. and Lempel, A., On MDS Codes via Cauchy Matrices, IEEE Trans. Inform. Theory, 1989, vol. 35, no. 6, pp. 1314–1319.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kéri, G., Types of Superregular Matrices and the Number of n-Arcs and Complete n-Arcs in PG(r, q), J. Combin. Des., 2006, vol. 14, no. 5, pp. 363–390; 2008, vol. 16, no. 3, pp. 262.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lidl, R. and Niederreiter, H., Finite Fields, Reading: Addison-Wesley, 1983. Translated under the title Konechnye polya, 2 vols., Moscow: Mir, 1988.zbMATHGoogle Scholar
  9. 9.
    Davydov A.A., Zyablov V.V., Kalimullin R.E. Subcodes of Reed-Solomon Code with Special Properties, in Proc. 12th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2010), Novosibirsk, Russia, 2010, pp. 116–122.Google Scholar
  10. 10.
    Djurdjevic I., Xu J., Abdel-Ghaffar K.A.S., Lin S. A Class of Low-Density Parity-Check Codes Constructed Based on Reed-Solomon Codes with Two Information Symbols, Proc. 15th Int. Sympos. on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-15), Toulouse, France, 2003, Fossorier, M.P.C., Høholdt, T., and Poli, A., Eds., Lect. Notes Comp. Sci., vol. 2643, Berlin: Springer, 2003, pp. 98–107.CrossRefGoogle Scholar
  11. 11.
    Bassalygo, L.A., Dodunekov, S.M., Zinoviev, V.A., and Helleseth, T., The Grey-Rankin Bound for Nonbinary Codes, Probl. Peredachi Inf., 2006, vol. 42, no. 3, pp. 37–44 [Probl. Inf. Trans. (Engl. Transl.), 2006, vol. 42, no. 3, pp. 197–203].Google Scholar
  12. 12.
    Tao, T. and Vu, V.H., Additive Combinatorics, New York: Cambridge Univ. Press, 2006.zbMATHCrossRefGoogle Scholar
  13. 13.
    Croot, E.S., III, and Lev, V.F., Open Problems in Additive Combinatorics, Additive Combinatorics, Granville, A., Nathanson, M.B., and Solymosi, J., Eds., CRM Proc. Lecture Notes, vol. 43, Providence: AMS, 2007, pp. 207–233.Google Scholar
  14. 14.
    Frankl, P. and Desa, M., On the Maximum Number of Permutations with Given Maximal or Minimal Distance, J. Combin. Theory, Ser. A, 1977, vol. 22, no. 3, pp. 352–360.zbMATHCrossRefGoogle Scholar
  15. 15.
    Blake, I.F. and Kith, K., On the Complete Weight Enumerator of Reed-Solomon Codes, SIAM J. Discrete Math., 1991, vol. 4, no. 2, pp. 164–171.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • A. A. Davydov
    • 1
    Email author
  • V. V. Zyablov
    • 1
  • R. E. Kalimullin
    • 1
  1. 1.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations