Related papers: Goppa goemetry codes via elementary methods (In Po…
In the past few years, substantial progress has been made in the understanding of the algebra of kappa classes on the moduli spaces of curves. My goal here is to provide a short introduction to the new results. Along the way, I will discuss…
A weighted Hamming metric is introduced in [4] and it showed that the binary generalized Goppa code is a perfect code in some weighted Hamming metric. In this paper, we study the weight structures which admit the binary Hamming code and the…
In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…
Lyon's rough paths give an algebraic and analytic framework for Stieltjes integrals in a regime of low regularity where the usual Riemann-Stieltjes integral does not converge. Before we may rigorously define rough paths, we start with the…
In [4] Camps-Moreno et al. treated (relative) generalized Hamming weights of codes from extended norm-trace curves and they gave examples of resulting good asymmetric quantum error-correcting codes employing information on the relative…
The paper contains a survey of the results obtained during the last ten years in the theory of elliptic boundary problems in H\"ormander function spaces, developed by the authors, and other related results of modern analysis. The basics of…
We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…
The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\"ottingen in 1854 entitled "\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie…
We study a connection between two topics: Decoding of Goppa codes arising from an algebraic curve, and rank two extensions of certain line bundles on the curve.
We define Convolutional Goppa Codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some Maximum-Distance Separable (MDS)…
This basic introduction to tropical geometry is hopefully accessible to a first years student in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking.…
We give a general method to construct MDS one-dimensional convolutional codes. Our method generalizes previous constructions of H. Gluesing-Luerssen and B. Langfeld. Moreover we give a classification of one-dimensional Convolutional Goppa…
In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this…
The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
This is the first paper in a series devoted to understanding the classical and quantum nature of edge modes and symmetries in gravitational systems. The goal of this analysis is to: i) achieve a clear understanding of how different…
The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized…
Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also…
In his G\'eom\'etrie (1637) Descartes introduces the algebra of segments. This is a fundamental step in the mathematical treatment of variable quantities before the creation of differential calculus. It is an algebra with symbols but…