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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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