On the gonality of curves, abundant codes and decoding

Abstract

Listen

Translate article

Let X be a curve defined over the finite field Fq with q elements. The genus of X is denoted by g(X ), or more often by g. Let P1, . . . , Pn be n distinct rational points on the curve X . Let D be the divisor P1 + . . . + Pn. Let G be a divisor on X of degree m. The code CL(D,G) is defined as the image of L(G) in F n q , under the evaluation map f 7−→ (f(P1), . . . , f(Pn)). Goppa [5] showed that the functional code CL(D,G) has dimension at least m + 1 − g and minimum distance at least n − m in case m 2g − 2, then the dimension is equal to m + 1 − g. We call n − m the Goppa designed minimum distance of CL(D,G), and denote it by dG. Tsfasman, Vladuţ, Zink and Ihara showed that modular curves have many rational points with respect to the genus, if q is a square, that is to say N ∼ ( √ q − 1)g, see [21, 4.1.52]. In case q ≥ 49 the Tsfasman-Vladuţ-Zink bound RTV Z gives an improvement of the Gilbert-Varshamov bound RGV , see [21, 3.4.4]. In the corners, where the graphs of RTV Z and RGV meet, Vladuţ made a slight improvement, moreover he showed that there are codes, comming from curves, with parameters lying on the maximum of the above mentioned bounds, see [21, 3.4.11]. Later Pellikaan, Shen and van Wee [14] proved that every linear code can be represented with a curve, but if one imposes the condition m < n, then long binary algebraic-geometric codes have information rate at most 1 2 . So the question of finding good codes can be restated in the question: Which divisors give good codes ? There are two ways to improve the bounds Goppa gave. In the first place by taking divisors such that the dimension is bigger than m + 1 − g. These are so called special divisors. If the field of constants is algebraically closed and g − k(g − m + k − 1) ≥ 0, then there exists a divisor of degree m and dimension at least k , by Brill-Noether theory. But this is no longer true over a finite field. The second possibility, which we will pursue in this paper, is to try to improve the bound on the minimum distance. In section 2 we show that the minimum distance is at least t(X ) − a for an abundant