Exponential sums and Goppa codes. I

Moreno, Carlos J.; Moreno, Oscar

doi:10.1090/S0002-9939-1991-1028291-1

Exponential sums and Goppa codes. I
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Proc. Amer. Math. Soc. 111 (1991), 523-531 Request permission

Abstract:

A bound is obtained which generalizes the Carlitz-Uchiyama result, based on a theorem of Bombieri and Weil about exponential sums. This new bound is used to estimate the covering radius of long binary Goppa codes. A new lower bound is also derived on the minimum distance of the dual of a binary Goppa code, similar to that for BCH codes. This is an example of the use of a number-theory bound for the problem of the estimation of minimum distance of codes, as posed in research problem 9.9 of Mac Williams and Sloane, The Theory of Error Correcting Codes.

References

Enrico Bombieri, On exponential sums in finite fields, Amer. J. Math. 88 (1966), 71–105. MR 200267, DOI 10.2307/2373048
L. Carlitz and S. Uchiyama, Bounds for exponential sums, Duke Math. J. 24 (1957), 37–41. MR 82517, DOI 10.1215/S0012-7094-57-02406-7
Philippe Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inform. Theory IT-21 (1975), no. 5, 575–576. MR 411819, DOI 10.1109/tit.1975.1055435
Tor Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Discrete Appl. Math. 11 (1985), no. 2, 157–173. MR 789141, DOI 10.1016/S0166-218X(85)80006-8

The theory of error correcting codes

Carlos Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics, vol. 97, Cambridge University Press, Cambridge, 1991. MR 1101140, DOI 10.1017/CBO9780511608766
A. Tietäväinen, On the covering radius of long binary BCH codes, Discrete Appl. Math. 16 (1987), no. 1, 75–77. MR 869739, DOI 10.1016/0166-218X(87)90055-2
André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. MR 27006, DOI 10.1073/pnas.34.5.204