A new effective decoding algorithm is presented for arbitrary algebraic-geometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard's algorithm, since the method corrects up to the half of the Goppa distance with complexity order O(n**2.81), and with no further assumption on the degree of the divisor G.
Power decoding is a partial decoding paradigm for arbitrary algebraic geometry codes for decoding beyond half the minimum distance, which usually returns the unique closest codeword, but in rare cases fails to return anything. The original version decodes roughly up to the Sudan radius, while an improved version decodes up to the Johnson radius, but has so far been described only for Reed--Solomon and one-point Hermitian codes. In this paper we show how the improved version can be applied to any algebraic geometry code.
Duality and decoding of linearized Algebraic Geometry codes
CNRS, Mathematical Institute of Bordeaux, University of Bordeaux（法国国家科学研究中心，波尔多大学数学研究所）；Mathematical Institute of Bordeaux, University of Bordeaux（波尔多大学数学研究所）
In this paper we present a decoding algorithm for algebraic geometry codes with error-correcting capacity beyond half the designed distance of the code. This algorithm comes as a fusion of the Power Error Locating Pairs algorithm for algebraic geometry codes and the technique used by Ehrhard in order to correct these codes up to half the designed distance. The decoding radius of this algorithm reaches that of Sudan algorithm, without any penalty given by the genus of the curve.
We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed--Solomon or algebraic geometry codes, the algorithm is a reformulation of the so--called {\em power decoding} algorithm. Asymptotically, it corrects errors up to Sudan's radius. In addition, this new framework applies to any code benefiting from an error locating pair. Similarly to Pellikaan's and Kötter's approach for unique algebraic decoding, our algorithm provides a unified point of view for decoding codes with an algebraic structure…
Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory. Under appropriate assumptions, general witness sets enable numerical algorithms such as sampling and membership. These assumptions hold for products of flag manifolds. We introduce Schubert witness sets, which provide general witness sets for Grassmannians …
In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.
Rank Decoding (RD) is the main underlying problem in rank-based cryptography. Based on this problem and quasi-cyclic versions of it, very efficient schemes have been proposed recently, such as those in the ROLLO and RQC submissions, which have reached the second round of the NIST Post-Quantum competition. Two main approaches have been studied to solve RD: combinatorial ones and algebraic ones. While the former has been studied extensively, a better understanding of the latter was recently obtained by Bardet et al. (EUROCRYPT20) where it appeared that algebraic attacks can often be more efficie…
We develop the idea of using an algebraic-geometry approach to classical differential geometry problems. Consider an orthogonal net constructed according to algebraic-geometric data we obtain a set of smooth orthogonal nets that are Ribaucour transformations of the initial orthogonal net.
We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.
Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.
In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.
This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.
