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Oggier F., Viterbo E. Algebraic Number Theory and Code Design for Rayleigh Fading Channels
Из серии Foundations and Trends in Communications and Information Theory издательства NOWPress, 2004, -97 pp.

Elementary number theory was the basis of the development of error correcting codes in the early years of coding theory. Finite fields were the key tool in the design of powerful binary codes and gradually entered in the general mathematical background of communications engineers. Thanks to the technological developments and increased processing power available in digital receivers, attention moved to the design of signal space codes in the framework of coded modulation systems. Here, the theory of Euclidean lattices became of great interest for the design of dense signal constellations well suited for transmission over the Additive White Gaussian Noise (AWGN) channel.
More recently, the incredible boom of wireless communications forced coding theorists to deal with fading channels. New code design criteria had to be considered in order to improve the poor performance of wireless transmission systems. The need for bandwidth-efficient coded modulation became even more important due to scarce availability of radio bands. Algebraic number theory was shown to be a very useful mathematical tool that enables the design of good coding schemes for fading channels.
These codes are constructed as multidimensional lattice signal sets (or constellations) with particular geometric properties. Most of the coding gain is obtained by introducing the so-called modulation diversity (or signal space diversity) in the signal set, which results in a particular type of bandwidth-efficient diversity technique.
Two approaches were proposed to construct high modulation diversity constellations. The first was based on the design of intrinsic high diversity algebraic lattices, obtained by applying the canonical embedding of an algebraic number field to its ring of integers. Only later it was realized that high modulation diversity could also be achieved by applying a particular rotation to a multidimensional QAM signal constellation in such a way that any two points achieve the maximum number of distinct components. Still, these rotations giving diversity can be designed using algebraic number theory.
An attractive feature of this diversity technique is that a significant improvement in error performance is obtained without requiring the use of any conventional channel coding. This can always be added later if required.
Finally, dealing with lattice constellations has also the key advantage that an efficient decoding algorithm is available, known as the Sphere Decoder.
Research on coded modulation schemes obtained from lattice constellations with high diversity began more than ten years ago, and extensive work has been done to improve the performance of these lattice codes. The goal of this work is to give both a unified point of view on the constructions obtained so far, and a tutorial on algebraic number theory methods useful for the design of algebraic lattice codes for the Rayleigh fading channel.

Introduction.
The Communication Problem.
Some Lattice Theory.
The Sphere Decoder.
First Concepts in Algebraic Number Theory.
Ideal Lattices.
Rotated Zn–lattices Codes.
Other Applications and Conclusions.
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