Fibre products of Kummer covers and curves with many points

We study the general fibre product of any two Kummer covers of the projective line over finite fields. Under some assumptions, we obtain an involved condition for the existence of rational points in the fibre product over a rational point of the projective line so that we determine the exact number of the rational points. Using this, we construct explicit examples of such fibre products with many rational points. In particular we obtain a record and a new entry for the table (http://www.science.uva.nl/^~geer/tables-mathcomp15.ps).      

Finite piecewise polynomial parametrization of plane rational algebraic curves

We present an algorithm with the following characteristics: given a real non-polynomial rational parametrization of a plane curve and a tolerance , is decomposed as union of finitely many intervals, and for each interval I of the partition, with the exception of some isolating intervals, the algorithm generates a polynomial parametrization . Moreover, as an option, one may also input a natural number N and then the algorithm returns polynomial parametrizations with degrees smaller or equal to N. In addition, we present an error analysis where we prove that the curve piece is in the offset region of at distance at most , and conversely. Authors partially supported by the Spanish “Ministerio de Educación y Ciencia” under the Project MTM2005-08690-C02-01, and by the “Dirección General de Universidades de la Consejería de Educación de la CAM y la Universidad de Alcalá” under the project CAM-UAH2005/053.     

Evaluation codes at singular points of algebraic differential equations

We use the special geometry of singular points of algebraic differential equations on the affine plane over finite fields to study the main features and parameters of error correcting codes giving by evaluating functions at sets of singular points. In particular, one gets new methods to construct codes with designed minimum distance. This work was partially supported by MCyT BFM2001-2251.     

Two applications of Weil-Serre's explicit formula

We apply Weil-Serre's explicit formula to produce non-trivial estimates of exponential sums along a curve and of the dual distance of subfield subcodes of algebraic-geometric (AG-)codes. These bounds work while the number of rational points of a curve is large when compared to its genus thus the Weil-Bombieri bound for exponential sums being trivial.Partly supported by Russian Fundamental Research Foundation (project N 93-012-458) and by the grant MPN000 from the International Science FoundationThe author is deeply grateful to Mme Aurélia Lozingot for her help in typing of the present paper.     

Weil bounds for singular curves

We give explicit bounds useful in estimating the number of points on a (possibly singular) space curve defined over a finite field. Our estimates involve the degrees of the polynomials defining the curve set-theoretically, and reduce to Weil's well-known estimate for nonsingular complete intersection curves.     

On the dimension of Goppa codes

A lower bound for the dimension of geometric BCH codes (i.e. subfield subcodes of Goppa codes) has been given by M. Wirtz [7]. We prove that this bound is actually exact for small enough divisorG.     

An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field

We establish a formula for the number of irreducible polynomialsf(x) over the binary fieldF ₂ of given degreen 2 for which the coefficient ofx ^n-1 and ofx is equal to 1. This formula shows that the number of such polynomials is positive for alln 2 withn 3. These polynomials can be applied in a construction of irreducible self-reciprocal polynomials overF ₂ of arbitrarily large degrees.