Algebraic curves over F2 with many rational points

Algebraic curves over F2 with many rational points

A smooth, projective, absolutely irreducible curve of genus 19 over F2 admitting an infinite S-class field tower is presented. Here S is a set of four F2-rational points on the curve. This is shown to imply that A(2) = limsup#X(F2)/g(X) ≥ 4/(19 − 1) ≈ 0.222. Here the limit is taken over curves X over F2 of genus g(X)→∞.

On Curves over Finite Fields with Many Rational Points

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field Fq2 whose number of Fq2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are Fq2 -isomorphic to y q + y = x for some m ∈ Z.

Nets, (t, S)-Sequences, and Algebraic Curves Over Finite Fields With Many Rational Points

Nets, (t; s)-sequences, and algebraic curves over nite elds with many rational points Nets and (t; s)-sequences are nite point sets, respectively innnite sequences, satisfying strong uniformity properties with regard to their distribution in the s-dimensional unit cube I s = 0; 1] s. The general theory of these com-binatorial objects was rst developed by the speaker 2], and they have attracted ...

Algebraic varieties with many rational points

sometimes we want to understand a typical equation, i.e., a general equation in some family. To draw inspiration (and techniques) from different branches of algebra it is necessary to consider solutions with values in other rings and fields, most importantly, finite fields Fq, finite extensions of Q, or the function fields Fp(t) and C(t). While there is a wealth of ad hoc elementary tricks to d...

Acquisition of Rational Points on Algebraic Curves