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Divisibility properties and new bounds for cyclic codes and exponential sums in one and several variables

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Abstract

Serre has obtained sharp estimates for the number of rational points on an algebraic curve over a finite field. In this paper we supplement his technique with divisibility properties for exponential sums to derive new bounds for exponential sums in one and several variables. The new bounds give us an improvement on previous bounds for the minimum distance of the duals of BCH codes. The divisibility properties also imply the existence of gaps in the weight distribution of certain cyclic codes, and in particular gives us that BCH codes are divisible (in the sense of H. N. Ward).

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References

  1. Serre, J. P.: Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris296 Série I, 397–402 (1983)

    Google Scholar 

  2. Carlitz, L., Uchiyama, S.: Bounds on exponential sums. Duke. Math. J.24, 37–41 (1957).

    Google Scholar 

  3. Weil, A.: On some exponential sums. Proc. N.A.S.34, 204–207 (1948)

    Google Scholar 

  4. Weil, A.: Courbes Algebriques et Varietes Abeliennes. Paris: Herman 1971

    Google Scholar 

  5. Bombieri, E.: On exponential sums in finite fields. Am. J. Math88, 71–105 (1966)

    Google Scholar 

  6. MacWilliams, F. J., Sloane, N. J. A.: The Theory of error correcting codes. Amsterdam: North-Holland Publishing Company 1977

    Google Scholar 

  7. Moreno, C. J.: Algebraic Curves over Finite Fields and Error Correcting Codes. Cambridge University Press 1991

  8. Moreno, C. J., Moreno, O.: Exponential Sums and Goppa Codes: I. Proc. of the AMS111 (2), 523–531 (1991)

    Google Scholar 

  9. Serre, J. P.: Nombres de points des courbes algebriques sur Fq. Seminaire de Théorie des Nombres de Bordeux22 (1982–83)

  10. Serre, J. P., Résumé des cours des 1983–1984. Annuaire du Collège de France (1984), pp. 79–83. Collected PapersIII. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  11. Deligne, P.: La conjecture de Weil. I. Publ. Math. I.H.E.S.48, 273–308 (1974)

    Google Scholar 

  12. Kumar, P. V., Moreno, O.: Prime-Phase Sequences with Periodic Correlation Properties Better Than Binary Sequences. IEEE Trans. Inform. Theory37:3, 603–616 (1991)

    Google Scholar 

  13. Joly, J. R.: Equations et variétés algébriques sur un corps fini. Enseignement Math. (2)19, 1–117(1973)

    Google Scholar 

  14. Moreno, O., Kumar, P. V.: Minimum distance bounds for cyclic codes and Deligne's Theorem. IEEE IT Trans. (in press)

  15. Ward, H. N.: Divisible codes. Arch. Math. (Basel)36, 485–494 (1981)

    Google Scholar 

  16. Ward, H. N.: A bound for divisible codes. IEEE IT Trans38 (1), 191–194 (1992)

    Google Scholar 

  17. Wolfmann, J.: New bounds on cyclic codes from algebraic curves. Lecture Notes in Computer Science vol. 388 pp. 47–62. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  18. Lachaud, G.: Artin-Schreier curves, exponential sums and the Carlitz-Uchiyama bound for geometric codes. J. Number Theory39 (1), 485–494 (1991)

    Google Scholar 

  19. Moreno, C. J., Moreno, O.: An improved Bombieri-Weil bound and applications to coding theory. J. Number Theory42, 32–46 (1992)

    Google Scholar 

  20. Adolphson, A., Speber, S.: p-adic estimates for exponential sums and the theorem of Chevalley-Wárning. Ann. Scient. E. N. Superior 4th series.20, 545–556 (1987)

    Google Scholar 

  21. Moreno, C. J., Moreno, O.: On the number of information symbols and covering radius of long Goppa codes. Proc. of Int. Workshop Algebraic and Combinat. Coding Theory. Varna, Bulgaria, Sept. 18–24 (1988)

  22. Moreno, O., Moreno, C. J.: A p-adic Serre bound. Preprint

  23. Andersón, D. R.: A new class of cyclic codes. SIAM J. Appl. Math.16, 181–197 (1968)

    Google Scholar 

  24. Mattson, H. F. Jr., Solomon, G.: A New Treatment of Bose-Chaudhuri Codes. J. Soc. Ind. Appl. Math.9, 654–669 (1961)

    Google Scholar 

  25. Moreno, O, Moreno, C. J.: Improvements of the Chevalley-Warning and the Ax-Katz Theorems. Am. J. Math. (in press)

  26. Moreno, O., Moreno, C. J.: An Elementary Proof of a Partial Improvement to the Ax-Katz Theòrem. Proceedings of Applied Algebra, Algebraic Algorithms and Error Correcting Codes (AAECC 10), Lecture Notes in Computer Science, vol.673, pp. 257–268. Berlin, Heidelberg, New York: Springer 1993

    Google Scholar 

  27. Moreno, O., Moreno, C. J.: The MacWilliams-Sloane Conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes, accepted in the IEEE IT Trans.

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Authors and Affiliations

  1. Department of Electrical Engineering - Systems, Tel-Aviv University, 69978, Ramat-Aviv, Israel

    S. Litsyn

  2. Baruch College and Graduate Center, City University of New York, Box 545, 10560, N. Saelm, NY, USA

    C. J. Moreno

  3. Department of Mathematics, University of Puerto Rico, P.R. 00931, Rio Piedras, USA

    O. Moreno

Authors
  1. S. Litsyn
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  2. C. J. Moreno
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  3. O. Moreno
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Additional information

The results of this paper were presented in the IEEE International Symposium on Information Theory, Budapest, Hungary, July 1991.

This work was partially supported by the Guastallo Fellowship and the Israeli Ministry of Science and Technology under Grant 5110431.

This work was partially supported by the National Science Foundation (NSF) under Grants DMS-8711566 and DMS-8712742.

This work was partially supported by NSF Grants RII-9014056, the Component IV of the EPSCoR of Puerto Rico Grant, and U.S. Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), of Cornell MSI. Contract DAAL03-91-C-0027.

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Litsyn, S., Moreno, C.J. & Moreno, O. Divisibility properties and new bounds for cyclic codes and exponential sums in one and several variables. AAECC 5, 105–116 (1994). https://doi.org/10.1007/BF01438279

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  • DOI: https://doi.org/10.1007/BF01438279

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