Algebraic geometric codes on certain plane curves

Abstract

Necessary and sufficient conditions are given for projective plane curves to have cusps isomorphic to the origin of an affine plane curve xa + yb. Based on them, a family of curves Cab having only one such cusp as a singular point and a family of curves rCab having only two such cusps as singular points are formulated. Also, the structures of algebraic geometric codes generated on curves contained in these families are shown, where a and b are relatively prime. Then the genus is (a − 1)/(b − 1)/2 and the basis of linear space L(m · P) is given by rational functions xiyi only. Second, by giving concrete examples, it is shown that families of curves Cab and rCab contain many curves which attain Hasse‐Weil upper bound.

On the existence of order functions

“…Proof. See also [5,15]. A generalization of this proposition will be given in Theorem 5.11 and Proposition 5.12.…”

Untitled

“…Let a, b ∈ ‫ޚ‬ ≥2 be coprime. A C ab curve is a curve having a rational place with Weierstrass semigroup a‫ޚ‬ ≥0 + b‫ޚ‬ ≥0 ; see [Miura 1992]. Any C ab curve is defined by a Weierstrass equation…”

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

“…In this paper we present an algorithm computing such a basis. An affine algebraic curve with one rational place Q at infinity is easy to handle and used extensively in the literature (Ganong, 1979;Porter, 1988;Miura, 1992Miura, , 1994Miura, , 1997Miura, , 1998Porter et al, 1992;Saints and Heegard, 1995). For a divisor D we define L(D + ∞Q) := ∞ i=1 L(D + iQ).…”