Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve

Abstract

Under the assumption that we have defining equations of an affine algebraic curve in special position with respect to a rational place Q, we propose an algorithm computing a basis of L(D) of a divisor D from an ideal basis of the ideal L(D + ∞Q) of the affine coordinate ring L(∞Q) of the given algebraic curve, where. Elements in the basis produced by our algorithm have pairwise distinct discrete valuations at Q, which is convenient in the construction of algebraic geometry codes. Our method is applicable to a curve embedded in an affine space of arbitrary dimension, and involves only the Gaussian elimination and the division of polynomials by the Gröbner basis for the ideal defining the curve.

Generalization of the Lee–OʼSullivan list decoding for one-point AG codes

“…By [13,Corollary 2.3] we have L(−D + ∞Q) = f . By [13,Corollary 2.5] we have L(−iD + ∞Q) = f i . Example 10 This is continuation of Example 2.…”

List decoding algorithm based on voting in Gröbner bases for general one-point AG codes

“…Define L(−G + ∞Q) = ∞ i=1 L(−G + iQ) for a positive divisor G of F/F q . Then L(−D + ∞Q) is an ideal of L(∞Q) (Matsumoto and Miura, 2000a). Let η i be any element in L(−D + ∞Q) such that lm(η i ) = x j 1 y i with j being the minimal given i.…”

“…Since we study list decoding, we cannot assume the original transmitted codeword nor the error vector as in Lee et al (2012). Nevertheless, the original theorems in Lee et al (2012) Matsumoto and Miura (2000a)).…”

Computing Riemann–Roch Spaces in Algebraic Function Fields and Related Topics

“…These solutions can roughly be divided into geometric and arithmetic methods due to their origin or background. The geometric methods (Le Brigand and Risler, 1988;Volcheck, 1994Volcheck, , 1995Haché, 1995;Huang and Ierardi, 1998) use the Brill-Noether method of adjoints (Brill and Noether, 1874;Noether, 1884) whereas the arithmetic methods (Coates, 1970;Davenport, 1981;Hess, 1999;Matsumoto and Miura, 2000) use a strategy involving ideals of integral closures, the basic idea of which essentially dates back to Dedekind and Weber (1882) (compare also Hensel and Landsberg, 1902). These methods usually deal with series expansions of algebraic functions at special places which results in a number of technical problems: assume, for example, that F/k is defined by some plane curve with a prescribed mapping to P 1 .…”

“…Solutions to this task have been considered in many places for important applications such as, for example, the construction of algebraic geometric codes (Le Brigand and Risler, 1988;Haché, 1995;Matsumoto and Miura, 2000), the explicit addition in the divisor class group (Volcheck, 1994(Volcheck, , 1995Huang and Ierardi, 1998), symbolic parametrizations of curves (van Hoeij, 1995(van Hoeij, , 1997, integration of algebraic functions (Davenport, 1981), the study of diophantine equations (Coates, 1970) or the computation of divisor class groups of global function fields and related problems (Hess, 1999). These solutions can roughly be divided into geometric and arithmetic methods due to their origin or background.…”