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Codes and character sums

  • A. Tietäväinen
Section I Coding And Algebraic Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 388)

Abstract

The proofs of the results of the sections 4 and 5 depend heavily on the deep works of Carlitz and Uchiyama, Deligne, Lang and Weil, and Bombieri, because the essential difficulty was that of estimating an exponential sum. It looks possible that in some cases this difficulty could be avoided by relating the demands of (8) and (9) to the estimate given by (10). This approach might lead to elementary proofs and at least it improves some bounds for acceptable n's. In fact, the bounds like (13) are not very precise, being merely given for convenience, and they could be improved at the cost of considerable complications. Of course it would be nice to find precise bounds.

As seen above, for the minimum distances and for the covering radii of binary BCH (and Goppa) codes we have found good asymptotic results. The nonbinary case is quite different: There is a large gap between the best known upper bounds and the best known lower bounds. It would be desirable to close that gap.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • A. Tietäväinen
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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