Omega: a system for the effective construction, coding and decoding of block error-correcting codes

L'objecte d'aquest treball és explicar com es poden implementar d'una manera efectiva, mitjançant el programa de manipulació simbòlica OMEGA, algunes de les construccions i operacions més importants de la teoria de codis correctors algebraics. Per als codis alternants, la classe més important que considerem, i que inclou els codis BCH, RS i de Goppa clàssics, presentem una implementació de l'algorisme de descodificació de Berlekamp-Massey. Per als codis cíclics, implementem l'algorisme de descodificació de Meggitt, i il·lustrem el seu funcionament, mitjançant la construcció de les corresponents taules de síndromes de Meggitt, per als codis de Golay. Finalment, presentem diversos altres grups de funcions, així com els càlculs i problemes (encara circumscrits gairebé a l'àrea de codis correctors) que ens permeten resoldre.In this work, we show how to implement effective constructions, coding and decoding of algebraic codes by means of Omega, a system specifically designed and programmed for general mathematical computations. For alternant codes, the main class we consider (which includes BCH, RS and classical Goppa codes), we give an implementation of the Euclidean division BM decoding algorithm. For cyclic codes we implement the Meggitt decoder, and to illustrate how it works we provide an implementation of the Meggitt syndrome tables for the two Golay codes. Finally, we present several other groups of functions and the computations and problems (still almost in the area of error-correcting codes) they solve

Sir Michael Atiyah : vida i obra

L'objecte d'aquest article és donar una perspectiva general de l'impacte científic i humà de Sir Michael Atiyah. La presentació de les principals idees i resultats amb què ha contribuït al coneixement matemàtic es complementa amb informacions sobre les persones que, per una raó o una altra, han tingut o tenen un paper rellevant en la seva trajectòria. El «racó matemàtic» (darrera secció) s'ha inclòs per facilitar, a les persones més motivades, una apreciació més detallada dels continguts matemàtics, i de física matemàtica, de l'obra d'Atiyah. Junt amb les referències bibliogràfiques, ens agradaria que aquest treball fos una invitació a explorar més profundament el seu lúcid pensament.The purpose of this paper is to provide a general perspective of the scientific and human impact of Sir Michael Atiyah. The presentation of the main ideas and results with which he has enriched mathematical knowledge is enhanced with information about people that have had or have a relevant role in his endeavors. The “mathematical corner” (the last section) is meant to facilitate, to the more motivated readers, a more detailed appreciation of the mathematics, and mathematical physics, in Atiyah’s works. Together with the bibliographical references, we would like that this article were an invitation to explore more deeply his lucid thinking

Computing the characteristic numbers of the variety of nodal plane cubics in P3

AbstractIn this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family Unod of non-degenerate nodal plane cubics in P3, first obtained by Schubert in his Kalkül der abzählenden Geometrie. The main geometric contribution is a detailed study of a variety Xnod, which is a compactification of the family Unod, including the boundary components (degenerations) and a generalization to P3 of a formula of Zeuthen for nodal cubics in P2. The computations have been carried out with the Wiris boost WIT

Real spinorial groups: a short mathematical introduction

This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry (or orthogonal geometry, for short) and how they relate to the group of isometries of that geometry. After a concise mathematical introduction, it offers an axiomatic presentation of the geometric algebra of an orthogonal geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index. Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students

Codis correctors d'errors i criptografia postquàntica

Revisitem el sistema criptogràfic de clau pública de McEliece, introduït fa quaranta anys, amb l'ajuda de recursos desenvolupats recentment: una millora del descodificador de Peterson-Gorenstein-Zierler per als codis correctors d'errors alternants; un sistema de computació simbòlica i un paquet d'utilitats funcionals per als càlculs emprats en la definició, codificació i descodificació de codis correctors d'errors, tot programat en Python, i una pàgina web que dona accés lliure als materials generats pel projecte. L'interès principal del sistema de McEliece rau en el fet que és un candidat seriós per a un estàndard de criptografia postquàntica.The forty-year old McEliece public-key crypto-system is revisited with the help of recently developed resources: an improved Peterson-Gorenstein-Zierler decoder for alternant error-correcting codes; a symbolic computation system and a package of functional utilities for the computations involved in defining, coding and decoding error-correcting codes, fully programmed in Python; and a web page with free-access to the materials generated by the project. The main interest of the McEliece system stems from it being a serious candidate for a post-quantum cryptography standard

A geometric algebra invitation to space-time physics, robotics and molecular geometry

This book offers a gentle introduction to key elements of Geometric Algebra, along with their applications in Physics, Robotics and Molecular Geometry. Major applications covered are the physics of space-time, including Maxwell electromagnetism and the Dirac equation; robotics, including formulations for the forward and inverse kinematics and an overview of the singularity problem for serial robots; and molecular geometry, with 3D-protein structure calculations using NMR data. The book is primarily intended for graduate students and advanced undergraduates in related fields, but can also benefit professionals in search of a pedagogical presentation of these subjects