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Die kompakte Darstellung einer in sich geschlossenen Theorie der linearen Codes wird vervollständigt durch die Implementierung eines Codes für AVR-Mikrocontroller. Zur Straffung der Entwicklung der Theorie wird etwas Homologie-Theorie eingesetzt. Es wird eine einfache Methode zur Konstruktion von Codes mit gegebenen Eigenschaften vorgestellt. Die Realisierung der Arithmetik endlicher Körper ist die Grundlage linearer Codes. Es werden deshalb zwei Verfahren hergeleitet und für verschiedene Mikrocontroller implementiert. Zur Konstruktion zyklischer Codes sind Polynome zu zerlegen, dazu werden zwei Verfahren ausführlich abgeleitet. Lineare Codes erfordern Polynomarithmetik und die Lösung linearer Gleichungssysteme über endlichen Körpern. Es wird gezeigt, wie beides in sehr effektive Programme für AVR-Mikrocontroller umgesetzt werden kann. Um zu einer durchgehend einheitlichen Symbolik zu gelangen enthält das Buch ein längeres Kapitel mit allen benötigten algebraischen Grundlagen. Weitere Hilfsmittel werden also nicht benötigt.

The common property among turbo-like code is that they consist of very simple constituent codes that are connected to each other with random or pseudorandom interleavers. The crucial novelty in these codes is the iterative decoding. This means that the constituent codes are decoded separately, which is ef?cient and practically feasible since they are very simple codes. Then, they pass new information to each other in a course of a few iterations. It has been shown that iterative decoding is a generalization of the well-known probability or belief propagation algorithm. The belief propagation algorithm that has been essential for development of new ideas throughout this work is described in the context of coding. The basic theorems for this algorithm are explained and proven in the following paragraphs. Thisis then followed by a description of the computational algorithm. The probability propagation algorithm is proven in c- junctionwithatree-structuredgraph–graphswithoutanycycle.Infact,thegraphical representation of any problem solved by this algorithm is the centerpiece of the algorithm. The generalization of the algorithm for graphs with cycles is presented later on. Representation of codes on graph is the next step towards characterization of the iterative decoding as an example of the probability propagation algorithm. The graph representations are presented for a few codes that are commonly used in turbo-like codes.

The fascinating theory of error-correcting codes is a rather new addition to the list of mathematical disciplines. It grew out of the need to communicate information electronically, and is currently no more than 60 years old. - ing an applied discipline by de?nition, a surprisingly large number of pure mathematical areas tie into Coding Theory. If one were to name just the most important connections, one would start of course with Linear Algebra, then list Algebra and Combinatorics, and further mention Number Theory and - ometry as well as Algebraic Geometry. Being a thorough introduction to the ?eld, this book starts from the very beginning, which is the channel model of communication in the presence of noise. From there, we develop the fundamental concepts of error-correcting codes, like the Hamming metric and the maximum likelihood decoding pr- ciple. After discussing dual codes and simple decoding procedures, this book takes an unusual turn. The standard approach would be to move on from there and introduce either more theory or present standard constructions of codes. The approach taken here is different.

This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

Find Codes, Ciphers and Secret Writing by Martin Gardner in Paperback and other formats in Reference > Questions & Answers.

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