Related papers: Coset bounds for algebraic geometric codes
Recent results of Cascudo, Cramer, and Xing on the construction of arithmetic secret sharing schemes are improved by using some new bounds on the torsion limits of algebraic function fields. Furthermore, new bounds on the torsion limits of…
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
We study the algebraic geometry of a family of evaluation codes from plane smooth curves defined over any field. In particular, we provide a cohomological characterization of their dual minimum distance. After having discussed some general…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\in\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This…
We give a lower bound for the Gaussian curvature of convex level sets of minimal graphs and the solutions to semilinear elliptic equations with the norm of boundary gradient and the Gaussian curvature of the boundary.
We consider complex projective schemes $X\subset\Bbb{P}^{r}$ defined by quadratic equations and satisfying a technical hypothesis on the fibres of the rational map associated to the linear system of quadrics defining $X$. Our assumption is…
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
We give an explicitly computable lower bound for the arithmetic self-intersection number of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In…
In this paper we compute the order (or Feng-Rao) bound on the minimum distance of one-point algebraic geometry codes, when the Weierstrass semigroup at the point Q is an Arf semigroup. The results developed to that purpose also provide the…
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex…
In this article, the minimum distance of the dual $C^{\bot}$ of a functional code $C$ on an arbitrary dimensional variety $X$ over a finite field $\F_q$ is studied. The approach consists in finding minimal configurations of points on $X$…
We give upper bounds on the genus of a curve with general moduli assuming that it can be embedded in a projective nonsingular surface $Y$ so that $\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate Kodaira…
In this paper we present several classes of asymptotically good concatenated quantum codes and derive lower bounds on the minimum distance and rate of the codes. We compare these bounds with the best-known bound of…
In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve…
The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this geometrical description is less trivial, it can be…
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the…
Algebraic-geometric codes on Garcia-Stichtenoth family of curves are used to construct the asymptotically good quantum codes.
We bound the size of $d$-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.
Let \({\mathbb K}\) be any field, let \(X\subset {\mathbb P}^{k-1}\) be a set of \(n\) distinct \({\mathbb K}\)-rational points, and let \(a\geq 1\) be an integer. In this paper we find lower bounds for the minimum distance \(d(X)_a\) of…