Related papers: How arithmetic and geometry make error correcting …
In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of…
Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
This is a comprehensive review on fault-tolerant topological quantum computation with the surface codes. The basic concepts and useful tools underlying fault-tolerant quantum computation, such as universal quantum computation, stabilizer…
In the first part of this article, we consider ruled surfaces defined over a finite field; we introduce invariants for them, and describe some explicit contructions that illustrate possible behaviour of these invariants. In the second part,…
This document contains the notes of a lecture I gave at the "Journ\'ees Nationales du Calcul Formel" (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts:…
These are notes of a talk, given at the 'arithmetique en plat pays'-meeting in february 2020, on the potential uses of geometries over the fake field $\mathbb{F}_1$ to zeta functions and counting measures on motives.
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
In the paper Factorisation of division polynomials (H. Verdure, Proc. japan Academy, Ser A. 80 n. 5), Verdure gives the factorisation patterns of division polynomials of elliptic curves defined over a finite field. However, the result given…
This is a renovated list of open problems, to appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu.
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
The following is an extended version of a talk given at the Kinosaki Symposium on Algebraic Geometry in October 2011. The aim is to give an overview of product-quotient surfaces, the results that have been proven so far in collaboration…
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key…
In this paper we give several methods to construct curves over finite fields with many points and illustrate this with examples of the results.
These lectures discuss recent advances on syzygies on algebraic curves, especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant Conjectures. The methods used are largely geometric and variational, with a special…
Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman…
Algebraic Geometric codes associated to a recently discovered class of maximal curves are investigated. As a result, some linear codes with better parameters with respect to the previously known ones are discovered, and 70 improvements on…
The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on curves and surfaces. More recently, the…
Whereas for a substantial part, Finite Geometry during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important…
We survey the notion and history of error-correcting codes and the algorithms needed to make them effective in information transmission. We then give some basic as well as more modern constructions of, and algorithms for, error-correcting…
This note presents explicit equations (up to birational equivalence over $\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over $\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard…