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Improvement of the Delsarte Bound for τ-Designs in Finite Polynomial Metric Spaces

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Cryptography and Coding (Cryptography and Coding 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2260))

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Abstract

In this paper the problem for the improvement of the Delsarte bound for τ -designs in finite polynomial metric spaces is investigated. First we distinguish the two cases of the Hamming and Johnson Q- polynomial metric spaces and give exact intervals, when the Delsarte bound is possible to be improved. Secondly, we derive new bounds for these cases. Analytical forms of the extremal polynomials of degree τ +2 for non-antipodal PMS and of degree τ +3 for antipodal PMS are given. The new bound is investigated in the following asymptotical process: in Hamming space when τ and n grow simultaneously to infinity in a proportional manner and in Johnson space when τ, w and n grow simultaneously to infinity in a proportional manner. In both cases, the new bound has better asymptotical behavior then the Delsarte bound.

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References

  1. J. Bierbrauer, K. Gopalakrishnan, D.R. Stinson, A note on the duality of linear programming bounds for orthogonal arrays and codes, Bulletin of the ICA, 22, (1998), 17–24.

    MATH  MathSciNet  Google Scholar 

  2. P.G. Boyvalenkov, D.P. Danev, On Linear Programming Bounds for the Codes in Polynomial Metric Spaces, Problems of Information Transmission 2, (1998), 108–120. (English translation from Problemy Peredachi Informatsii).

    MathSciNet  Google Scholar 

  3. P. Delsarte, An Algebraic Approach to Association Schemes in Coding Theory, Philips Research Reports Suppl. 10, 1973.

    Google Scholar 

  4. P. Delsarte, Application and Generalization of the MacWilliams transform in Coding Theory, Proc. 15th Symp. on Inform. Theory in Benelux, Louvain-la-Neuve, Belgium, (1994), 9–44.

    Google Scholar 

  5. P. Delsarte, J.M. Goethals, J.J. Seidel, Spherical codes and designs, Geometricae Dedicata 6, (1977), 363–388.

    Article  MATH  MathSciNet  Google Scholar 

  6. C.F. Dunkl, Discrete quadrature and bounds on t-designs, Mich. Math. J. 26, (1979), 81–102.

    Article  MATH  MathSciNet  Google Scholar 

  7. R.W. Hamming, Error detecting and error correcting codes, Bell Syst. Tech. J., 29, (1950), 147–160.

    MathSciNet  Google Scholar 

  8. S.G. Hoggar, t-designs in projective spaces, Europ. J. Combin. 3, (1982), 233–254.

    MATH  MathSciNet  Google Scholar 

  9. V.I. Levenshtein, On choosing polynomials to obtain bounds in packing problems, In Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission Part II, Moscow, Vilnus, (1978), 103–108 (in Russian).

    Google Scholar 

  10. V.I. Levenshtein, Designs as Maximum Codes in Polynomial Metric Spaces, Acta Applicandae Mathematicae 29, (1992), 1–82.

    Article  MathSciNet  MATH  Google Scholar 

  11. V.I. Levenshtein, Packing and Decomposition Problems for Polynomial Association Schemes, Europ. J. Combinatorics 14, (1993), 461–477.

    Article  MATH  MathSciNet  Google Scholar 

  12. V.I. Levenshtein, Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces, IEEE Transactions on Information Theory 41(5), (1995), 1303–1321.

    Article  MATH  MathSciNet  Google Scholar 

  13. V.I. Levenshtein, On designs in compact metric spaces and a universal bound on their size, Discrete Mathematics 192, (1998), 251–271.

    Article  MATH  MathSciNet  Google Scholar 

  14. V.I. Levenshtein, Universal bounds for codes and designs, Chapter 6 in Handbook of Coding Theory, ed. V. Pless and W.C. Huffman, 1998, Elsevier Science B.V., 449–648.

    Google Scholar 

  15. V.I. Levenshtein, Equivalence of Delsarte’s bounds for codes and designs in symmetric association schemes, and some applications, Discrete Mathematics 197/198, (1999), 515–536.

    MathSciNet  Google Scholar 

  16. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977.

    Google Scholar 

  17. R.J. McElliece, E.R. Rodemich, H. Rumsey and L.R. Welch, New Upper Bounds on the rate of a code via the Delsarte-MacWilliams Inequalities, IEEE Transactions on Information Theory 23 (2), (1977), 157–166.

    Article  Google Scholar 

  18. S.I. Nikova, V.S. Nikov, Improvement of the Delsarte bound for τ-designs when it is not the best bound possible, submitted in Designs Codes and Cryptography.

    Google Scholar 

  19. C.R. Rao, Factorial experiments derivable from combinatorial arrangements of arrays, J.Roy. Stat. Soc. 89, (1947), 128–139.

    Google Scholar 

  20. D.K. Ray-Chaudhuri, R.M. Wilson, On t-designs, Osaka J. Math. 12, (1975), 737–744.

    MATH  MathSciNet  Google Scholar 

  21. R.C. Singleton, Maximum distance q-ary codes, IEEE Trans. Inform. Theory 10, (1964), 116–118.

    Article  MATH  MathSciNet  Google Scholar 

  22. V.M. Sidel’nikov, Extremal polynomials used in bounds of code volume, Problemy Peredachi Informatsii 16 (3), (1980), 17–30 (in Russian). English translation in Problems on Information Transmission 16 (3), (1980), 174-186.

    MathSciNet  Google Scholar 

  23. I. Schoeneberg, G. Szegö, An extremum problem for polynomials, Composito Math. 14, (1960), 260–268.

    Google Scholar 

  24. G. Szegö, Orthogonal polynomials, AMS Col. Publ., vol.23, Providence, RI, 1939.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department Electrical Engineering, ESAT/COSIC, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, B-3001, Heverlee-Leuven, Belgium

    Svetla Nikova

  2. Department of Mathematics and Informatics, Veliko Turnovo University, 5000, Veliko Turnovo, Bulgaria

    Ventzislav Nikov

Authors
  1. Svetla Nikova
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  2. Ventzislav Nikov
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Editor information

Editors and Affiliations

  1. Faculty of Applied Sciences, Department of Communication Systems, Lancaster University, LA1 4YR, Lancaster, UK

    Bahram Honary

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© 2001 Springer-Verlag Berlin Heidelberg

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Nikova, S., Nikov, V. (2001). Improvement of the Delsarte Bound for τ-Designs in Finite Polynomial Metric Spaces. In: Honary, B. (eds) Cryptography and Coding. Cryptography and Coding 2001. Lecture Notes in Computer Science, vol 2260. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45325-3_18

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  • DOI: https://doi.org/10.1007/3-540-45325-3_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43026-1

  • Online ISBN: 978-3-540-45325-3

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