arXiv subjects
Publications and source records attributed to Fernando Torres.
For a linear code $\cC$ of length $n$ and dimension $k$, Wolf noticed that the trellis state complexity $s(\cC)$ of $\cC$ is upper bounded by $w(\cC):=\min(k,n-k)$. In this paper we point out some new lower bounds for $s(\cC)$. In particular, if $\cC$ is an Algebraic Geometric code, then $s(\cC)\geq w(\cC)-(g-a)$, where $g$ is the genus of the underlying curve and $a$ is the abundance of the code.
We show that a F_{q^2}-maximal curve of genus q(q-3)/6 in characteristic three is either a non-reflexive space curve of degree q+1, or it is uniquely determined up to F_{q^2}-isomorphism by a plane model of Artin-Schreier type
We present predictions for numerous statistics related to the presence of voids in the distribution of galaxies in a cold dark matter model of structure formation using a semi-analytic model of galaxy formation. Our study is able to probe galaxies with masses as low as 10^9Msun/h corresponding to absolute magnitudes of M_bJ-5log(h)=-18.1 and M_r-5log(h)=-18.7. We quantify the void and underdense probability functions, distributions of nearest neighbour distances and void sizes and compute the density profiles of voids. These results are contrasted with the expectations for dark matter (and the difference examined in terms of the galaxy/dark matter biasing relation) and are compared to analytic predictions and observational data where available. The predicted void probability functions are consistent with those measured from the Center for Astrophysics redshift surveys given the rather large uncertainties in this relatively small (for studies of voids) observational sample. The size of the observational sample is too small to probe the bias between galaxies and dark matter that we predict. We also examine the predicted properties of galaxies living within voids and contrast these with the general galaxy population. Our predictions are aimed at forthcoming large galaxy redshift surveys which should for the first time provide statistically accurate measures of the void population.
Inspired by a recent paper of Garcia, Stichtenoth and Xing [2000, Compositio Math. 120, 137--170], we investigate the quotient curves of the Deligne-Lusztig curve associated to the Suzuki group.
We discuss sufficient conditions for a given curve to be covered by a maximal curve with the covering being unramified; it turns out that the given curve itself will be also maximal. We relate our main result to the question of whether or not a maximal curve is covered by the Hermitian curve. We also provide examples illustrating the results.
This is an expository paper concerning topics on rational points of curves defined over finite fields based on a paper by St"ohr and Voloch.
Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/3} is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume \dim(\cD_\cX)=5, and char(\fq)\geq 5. For q\equiv 1\pmod{4}, q\geq 17, the Fermat curve of equation x^{(q+1)/2}+y^{(q+1)/2}+1=0 is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-3)/8. For q\equiv 3\pmod{4}, q\ge 19, there are exactly two F_{q^2}-maximal curves of genus g=(q-1)(q-3)/8, namely the above Fermat curve and the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/4}. The above results provide some new evidences on maximal curves in connection with Castelnuovo's bound and Halphen's theorem, especially with extremal curves; see for instance the conjecture stated in Introduction.
Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5. This upper bound turns out to be sharp for q odd. We also study the maximality of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family of curves.
Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field F of order $q^2$. If the number of F-rational points of X satisfies the Hasse-Weil upper bound, then X is said to be F-maximal. For a point P_0\in X(F), let πbe the morphism arising from the linear series D:=|(q+1)P_0|, and let N:=dim(D). It is known that N\ge 2 and that πis independent of P_0 whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then π:X\to π(X) is a F-isomorphism. The non-singular model π(X) has degree q+1 and lies on a Hermitian variety defined over F of P^N(\bar F); Theorem 0.2: If X is F-maximal, then it is F-isomorphic to a curve Y in P^M(\bar F), with 2\le M\le N, such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of ¶^M(\bar F). Furthermore, Aut_F(X) is isomorphic to a subgroup of the projective unitary group PGU(M+1,q^2); Theorem 0.3: If X is F-birational to a curve Y embedded in P^M(\bar F) such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of P^M(\bar F), then X is F-maximal and X is F-isomorphic to Y.
An approach for the computation of upper bounds on the size of large complete arcs is presented. We obtain in particular geometrical properties of irreducible envelopes associated to a second largest complete arc provided that the order of the underlying field is large enough. We use Stoehr-Voloch's approach to the Hasse-Weil bound for rational points of curves defined over finite fields
The genus g of an F_{q^2}-maximal curve satisfies g=g_1:=q(q-1)/2 or g\le g_2:= [(q-1)^2/4]. Previously, such curves with g=g_1 or g=g_2, q odd, have been characterized up to isomorphism. Here it is shown that an F_{q^2}-maximal curve with genus g_2, q even, is F_{q^2}-isomorphic to the nonsingular model of the plane curve \sum_{i=1}^{t}y^{q/2^i}=x^{q+1}, q=2^t, provided that q/2 is a Weierstrass non-gap at some point of the curve.
We study geometrical properties of maximal curves having classical Weierstrass gaps.
We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.
We give a characterization of the Deligne-Lusztig curve associated to the Suzuki group based on the genus and the number of rational points of the curve.
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 $m \in Z^+$. As a consequence we show that a maximal curve of genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.
We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double covering of curves of genus large enough.
We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.
We extend results on Weierstrass semigroups at ramified points of double covering of curves to any numerical semigroup whose genus is large enough. As an application we strengthen the properties concerning Weierstrass weights in \cited{To}.