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.

PREFACE The increasing demand on high data rate and quality of service in wireless communication has to cope with limited bandwidth and energy resources. More than 50 years ago, Shannon has paved the way to optimal usage of bandwidth and energy resources by bounding the spectral efficiency vs. signal to noise ratio trade-off. However, as any information theorist, Shannon told us what is the best we can do but not how to do it [1]. In this view, turbo codes are like a dream come true: they allow approaching the theoretical Shannon capacity limit very closely. However, for the designer who wants to implement these codes, at first sight they appear to be a nightmare. We came a huge step closer in striving the theoretical limit, but see the historical axiom repeated on a different scale: we know we can achieve excellent performance with turbo codes, but not how to realize this in real devices.

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"The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. This book is about an example of such a connection: the relation between codes and lattices. Lattices are studied in number theory and in the geometry of numbers. Many problems about codes have their counterpart in problems about lattices and sphere packings. We give a detailed introduction to these relations including recent results of G. van der Geer and F. Hirzebruch. Let us explain the history of this book. In [LPS82] J. S. Leon, V. Pless, and N. J. A. Sloane considered the Lee weight enumerators of self-dual codes over the prime field of characteristic 5. They wrote in the introduction to their paper: ""The weight enumerator of anyone of the codes . . . is strongly constrained: it must be invariant under a three-dimensional representation of the icosahedral group. These invariants were already known to Felix Klein, and the consequences for coding theory were discovered by Gleason and Pierce (and independently by the third author) . . . (It is worth mentioning that precisely the same invariants have recently been studied by Hirzebruch in connection with cusps of the Hilbert modular surface associated with Q( J5)."