New curves with many points over small finite fields
نویسندگان
چکیده
منابع مشابه
Curves over Finite Fields with Many Points: an Introduction
The number of points on a curve defined over a finite field is bounded as a function of its genus g. In this introductory article, we survey what is known about the maximum number of points on a curve of genus g defined over Fq, including an exposition of upper bounds, lower bounds, known values of this maximum, and briefly indicate some methods of constructing curves with many points, providin...
متن کاملOn 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.
متن کاملHow to Construct Curves Over Finite Fields With Many Points
This bound was proved for elliptic curves by Hasse in 1933. Ever since, the question of the maximum number Nq(g) of points on an irreducible curve of genus g over a finite field of cardinality q could have been investigated. But for a long time it attracted no attention and it was only after Goppa introduced geometric codes in 1980 that this question aroused systematic attention, cf. [G], [M], ...
متن کاملRational Points on Curves over Finite Fields
Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...
متن کاملCounting Points on Curves over Finite Fields
Stanford University) Abstract: A curve is a one dimensional space cut out by polynomial equations, such as y2=x3+x. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables (x and y in the example) in the finite field that satisfy the given polyn...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Finite Fields and Their Applications
سال: 2013
ISSN: 1071-5797
DOI: 10.1016/j.ffa.2013.01.007