Related papers: Construction of Rational Surfaces Yielding Good Co…

We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the…

Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several…

We study the functional codes $C_2(X)$ defined on projective varieties $X$, in the case where $X\subset \mathbb{P}^3$ is a 1-degenerate quadric or a non-degenerate quadric (hyperbolic or elliptic). We find the minimum distance of these…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum…

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming…

Interest in the hulls of linear codes has been growing rapidly. More is known when the inner product is Euclidean than Hermitian. A shift to the latter is gaining traction. The focus is on a code whose Hermitian hull dimension and dual…

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem…

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC…

Surface codes are a promising method of quantum error correction and the basis of many proposed quantum computation implementations. However, their efficient decoding is still not fully explored. Recently, approaches based on machine…

In this paper, we give an effective and efficient algorithm which on input takes non-zero integers $A$ and $B$ and on output produces the generators of the Mordell-Weil group of the elliptic curve over $\mathbb{Q}(t)$ given by an equation…

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

In the article we construct low-rate non-split toric $q$-ary codes on some singular surfaces. More precisely, we consider non-split toric cubic and quartic del Pezzo surfaces, whose singular points are $\mathbb{F}_{\!q}$-conjugate. Our…

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…

This paper is a general survey of literature on Goppa-type codes from higher dimensional algebraic varieties. The construction and several techniques for estimating the minimum distance are described first. Codes from various classes of…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…