Locally Recoverable codes from rational maps

We give a method to construct Locally Recoverable Error-Correcting codes. This method is based on the use of rational maps between affine spaces. The recovery of erasures is carried out by Lagrangian interpolation in general and simply by one addition in some good cases.Comment: 19 page

Explicit Construction of AG Codes from Generalized Hermitian Curves

We present multi-point algebraic geometric codes overstepping the Gilbert-Varshamov bound. The construction is based on the generalized Hermitian curve introduced by A. Bassa, P. Beelen, A. Garcia, and H. Stichtenoth. These codes are described in detail by constrcting a generator matrix. It turns out that these codes have nice properties similar to those of Hermitian codes. It is shown that the duals are also such codes and an explicit formula is given.Comment: 13 pages, 3 figure

Nonbinary Quantum Codes from Two-Point Divisors on Hermitian Curves

Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum error-correction, the observed presence of asymmetry in some quantum channels led to the study of asymmetric quantum codes (AQECCs) where we no longer assume that the different types of errors are equiprobable. This paper considers quantum codes constructed from the two-point codes. In the asymmetric case, we show strict improvements over all possible finite fields for a range of designed distances. We produce large dimension pure AQECC and small dimension impure AQECC that have better parameters than AQECC from one-point codes. Numerical results for the Hermitian curves over F16 and F64 are used to illustrate the gain

The Deligne-Lusztig curve associated to the Suzuki group

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.Comment: LaTex 2e, 12 page

On a F_{q^2}-maximal curve of genus q(q-3)/6

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 typeComment: 11 pages, LaTeX, available at http://www.ime.unicamp.br/~ftorres/articles.html; We correct several misprints: one of them was the equation of the curve whose unicity was claimed; we also give a more detailed proof of the main resul

Codes defined by forms of degree 2 on hermitian surfaces and S\o rensen's conjecture

We study the functional codes $C_h(X)$ defined by G. Lachaud in $\lbrack 10 \rbrack$ where $X \subset {\mathbb{P}}^N$ is an algebraic projective variety of degree $d$ and dimension $m$. When $X$ is a hermitian surface in $PG(3,q)$, S{\o}rensen in \lbrack 15\rbrack, has conjectured for $h\le t$ (where $q=t^2$) the following result : # X_{Z(f)}(\mathbb{F}_{q}) \le h(t^{3}+ t^{2}-t)+t+1 which should give the exact value of the minimum distance of the functional code $C_h(X)$. In this paper we resolve the conjecture of S{\o}rensen in the case of quadrics (i.e. $h=2$), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weightComment: accepted for publication in Finite Fields and Their Application

Algebraic Geometric codes from Kummer Extensions

For Kummer extensions defined by $y^m = f (x)$, where $f (x)$ is a separable polynomial over the finite field $\mathbb{F}_q$, we compute the number of Weierstrass gaps at two totally ramified places. For many totally ramified places we give a criterion to find pure gaps at these points and present families of pure gaps. We then apply our results to construct many points algebraic geometric codes with good parameters.Comment: 16 page

Remarks on plane maximal curves

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.Comment: 14 pages, LaTex2

Improved Two-Point Codes on Hermitian Curves

One-point codes on the Hermitian curve produce long codes with excellent parameters. Feng and Rao introduced a modified construction that improves the parameters while still using one-point divisors. A separate improvement of the parameters was introduced by Matthews considering the classical construction but with two-point divisors. Those two approaches are combined to describe an elementary construction of two-point improved codes. Upon analysis of their minimum distance and redundancy, it is observed that they improve on the previous constructions for a large range of designed distances

A note on the genus of certain curves over finite fields

We prove the following result which was conjectured by Stichtenoth and Xing: let $g$ be the genus of a projective, irreducible non-singular curve over the finite field $\Bbb F_{q^2}$ and whose number of $\Bbb F_{q^2}$-rational points attains the Hasse-Weil bound; then either $4g\le (q-1)^2$ or $2g=(q-1)q$.Comment: 4 pages, Latex. Reason for resubmission: The proof of the theorem in the previous version of this paper was incomplet