Weierstrass pairs and minimum distance of Goppa codes (Q5929714)

As \textit{V. D. Goppa} himself observed (pp. 139--145 in [Geometry and Codes, Math. Appl., Kluwer Acad. Publishers (1988; Zbl 1097.14502)]), a good selection for the degree of the space of sections that defines a one-point code may imply an improvement of the usual Goppa lower bound for the minimum distance of the code. This was formalized by \textit{A. Garcia} and \textit{R. Lax} [Coding theory and algebraic geometry, Lect. Notes Math. 1518, 33--42 (1992; Zbl 0768.94023)], and by \textit{A. Garcia, S. J. Kim}, and \textit{R. F. Lax} [J. Pure Appl. Algebra 84, 199--207 (1993; Zbl 0768.94014)]. The present author studies two-point Goppa codes, namely those whose linear sections are defined by divisors of type \(aP+bQ\), where \(a\) and \(b\) are positive integers, and \(P\) and \(Q\) are Weierstrass points of the underlying curve. She takes advantage of several arithmetical properties of the so-called Weierstrass semigroup \(H(P,Q)\) of the pairs of points \(P\) and \(Q\) [see \textit{S. J. Kim}, Arch. Math. 62, No. 4, 73--82 (1994; Zbl 0815.14020); \textit{M. Homma}, Arch. Math. 67, 337--348 (1996; Zbl 0869.14015)] in order to generalize the aforementioned result of Garcia, Lax, and Kim to the case of two-point codes. Finally, the author specializes her results to the case of two-point Goppa codes on Hermitian curves. As a matter of fact, one-point Goppa codes on such curves has been intensively studied by several authors. For example \textit{H. Stichtenoth} [IEEE Trans. Inf. Theory IT-34, 1345-1348 (1989; Zbl 0665.94015)], \textit{P. Kumar} and \textit{K. Yang} [Coding theory and algebraic geometry, Lect. Notes Math. 1518, 99--107 (1992; Zbl 0763.94023)] computed the precise value for the minimum distance of such codes. Noticing that (for a fixed curve) the length of two-point Goppa codes is at most one less than the length of one-point Goppa codes, the current author constructs many examples where working with two-point Goppa codes provides better codes than those one can find with one-point codes.