On some open problems on maximal curves (original) (raw)
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On maximal curves with Frobenius dimension 3
Designs, Codes and Cryptography, 2009
Frobenius dimension is one of the most important birational invariants of maximal curves. In this paper, a characterization of maximal curves with Frobenius dimension equal to 3 is provided. Our main tool is the Natural Embedding Theorem for maximal curves. As an application, maximal curves with Frobenius dimension 3 defined over the fields with 16 and 25 elements are completely classified.
Further examples of maximal curves
Journal of Pure and Applied Algebra, 2009
It is shown that the curve y q 2 − y = x q n +1 q+1 over F q 2n with n ≥ 3 odd, that generalizes Serre's curve y 4 + y = x 3 over F 64 , is also maximal. We also investigate a family of maximal curves over F q 2n and provide isomorphisms between these curves.
1998
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod
On additive polynomials and certain maximal curves
Journal of Pure and Applied Algebra, 2008
We show that a maximal curve over F q 2 given by an equation A(X) = F(Y), where A(X) ∈ F q 2 [X] is additive and separable and where F(Y) ∈ F q 2 [Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F q 2. In the particular case where F(Y) = Y m , we show that the degree m is a divisor of q + 1.
The maximal rank conjecture for non-special curves in ? n
Mathematische Zeitschrift, 1987
In this paper we conclude our study ([2, 3]) about the postulation of "general" curves embedded in a projective space by a non-special linear system. Recall that a curve CcP" is said to be of maximal rank if for every k> 1, the natural map of restriction rc,,(k): Hoop ", @,(k))~H~ (gc(k)) is surjective or injective.
On maximal curves in characteristic two
Manuscripta Mathematica, 1999
The genus g of an q-maximal curve satisfies g=g 1≔q(q−1)/2 or . Previously, q-maximal curves with g=g 1 or g=g 2, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g 2, q even, is q-isomorphic to the non-singular model of the plane curve ∑i =1}t y q /2i =x q +1, q=2t , provided that q/2 is a Weierstrass non-gap at some point of the curve.