Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity A(), for = p n with p prime and n > 3 odd. A modular interpretation of the towers is given as well.
We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraic-geometric codes defined using a multiple of the point. A q-ary linear code of length n and dimension k is a vector subspace of dimension k of F n q , where F q… (More)
We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m , for some m ∈ Z +. As a consequence we show… (More)
The conference " Coding Theory " intended to be a platform where the rather inhomo-geneous coding community could exchange ideas. Both the engineers could present their mathematical problems and the mathematicians could report their progress. The result was a lively meeting with an open exchange of ideas and lots of discussion. Madhu Sudan presented his… (More)
Generalizing the norm and trace mappings for % O P /% O , we introduce an interesting class of polynomials over "nite "elds and study their properties. These polynomials are then used to construct curves over "nite "elds with many rational points. 1999 Academic Press
We study geometrical properties of maximal curves having classical Weierstrass gaps.
We present a new explicit tower of function fields (F n) n≥0 over the finite field with = q 3 elements, where the limit of the ratios (number of rational places of F n)/(genus of F n) is bigger or equal to 2(q 2 − 1)/(q + 2). This tower contains as a subtower the tower which was introduced by Bezerra– Garcia–Stichtenoth (see ), and in the particular case… (More)
Let C be a (non-singular, projective, geometrically irreducible, algebraic) curve of genus g defined over a finite field F q with q elements. We know after A. Weil that the number of F q-points of a curve of genus g defined over F q satisfies the following limitations: q + 1 − 2g √ q ≤ #C(F q) ≤ 1 + q + 2g √ q, where C(F q) denotes the set of F q-rational… (More)
Towers of function fields (resp., of algebraic curves) with positive limit provide examples of curves with large genus having many rational points over a finite field. It is in general a difficult task to calculate the genus of a wild tower. In this paper, we present a method for calculating the genus of certain Artin–Schreier towers. As an illustration of… (More)