Exponential sums and Goppa codes. I. (English) Zbl 0716.94010

Moreno, Carlos J.; Moreno, Oscar

Proc. Am. Math. Soc. 111, No. 2, 523-531 (1991).

Summary: 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 F. J. MacWilliams and N. J. A. Sloane, The theory of error correcting codes (Amsterdam 1977; Zbl 0369.94008).

MSC:

94B05	Linear codes (general theory)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11T23	Exponential sums
94B65	Bounds on codes

Keywords:

Goppa polynomial; exponential sums; covering radius of long binary Goppa codes; minimum distance of the dual of a binary Goppa; BCH codes

Citations:

Zbl 0369.94008