Consecutive Weierstrass Gaps and Minimum Distance of Goppa Codes

We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraic-geometric codes defined using a multiple of the point. A q-ary linear code of length n and dimension k is a vector subspace of dimension k of Fq , where Fq denotes the finite field with q elements. The minimum distance of a code is the minimum number of places in which two distinct codewords differ. The greater the minimum distance, the greater the number of errors that the code can detect or correct. For a linear code, the minimum distance is also the minimum weight of a nonzero codeword, where the weight of a codeword is the number of nonzero places in that codeword. A linear code of length n, dimension k and minimum distance d is called an [n, k, d]-code. V.D. Goppa [3,4] realized that one could use the Riemann-Roch Theorem to show that certain codes produced from two divisors G and D on a curve have good properties. In particular, he gave lower bounds for the minimum distances of these codes. In a previous article [1], the first and third authors showed that if G is taken to be a multiple of a point P , then knowledge of the gaps at P may allow one to say that the minimum distance of the resulting code is greater than Goppa’s lower bound. We also showed that the presence of t consecutive gaps, together with certain conditions on osculating spaces at the point P , could allow one to conclude that the resulting code has minimum distance at least t greater than Goppa’s bound. Here, we drop any assumptions about osculating spaces at P and show that, by assuming more about the gaps at P , one may obtain a similar result relating consecutive gaps with an improvement of Goppa’s bound on the minimum distance. In the first section, we give the necessary definitions. The second section contains our main results (Theorems 3 and 4), and in the final section, we present two examples illustrating the theorems. 1. Let X denote a nonsingular, geometrically irreducible, projective curve of genus g > 1 defined over Fq. Assume that X has Fq-rational points. Let D be a divisor on X defined over Fq (i.e., D is invariant under Gal(Fq/Fq)). Then L(D) will denote the Fq-vector space of all rational functions f on X , defined over Fq, with divisor (f) ≥ −D, together with the zero function, and Ω(D) will denote the Fq-vector space of all rational differentials η on X , defined over Fq, with divisor (η) ≥ D, together with the zero differential. Put l(D) = dimFqL(D) and i(D) = dimFqΩ(D). The Riemann-Roch Theorem states that l(D) = degD + 1− g + i(D) = degD + 1− g + l(K −D), 1 Partially supported by a CNPq grant 2 Partially supported by a KOSEF grant 3 Partially supported by NSF grant INT-9101337