## Algebraic Number Theory and Code Design for Rayleigh Fading ChannelsAlgebraic number theory is gaining an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems. Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been developed in the last ten years and many explicit code constructions based on algebraic number theory are now available. Algebraic Number Theory and Code Design for Rayleigh Fading Channels provides an overview of algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field are illustrated by many examples and by the use of computer algebra freeware in order to make it more accessible to a large audience. This makes the book suitable for use by students and researchers in both mathematics and communications. |

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### Contents

Introduction | 1 |

Some Lattice Theory | 15 |

First Concepts in Algebraic Number Theory | 37 |

Ideal Lattices | 57 |

Other Applications and Conclusions | 79 |

### Common terms and phrases

algebraic element algebraic integer algebraic lattice algebraic number field algebraic number theory AWGN basis is given bit labeling called canonical embedding chapter codewords coding gain complex lattice component interleaver compute the embeddings consider cyclotomic construction cyclotomic field Definition denote det(A diversity order dp,mjn embeddings kash example extension of Q field extension field of degree full-rank lattice fundamental parallelotope Gaussian channel given lattice Gram matrix ideal lattice ideal of OK integer components integral basis kash J C OK KASH/KANT lattice constellations lattice decoding algorithm lattice generator matrix lattice sphere packing matrix G maximal diversity minimal polynomial minimum product distance modulation diversity norm obtained Odlyzko's bound operation optimal OrderMaximal principal ideal Q(Cp quotient group radius Rayleigh fading channel real algebraic number real number fields Recall received point ring of integers shown signal constellations signal set Space-Time Block Codes Sphere Decoder sphere packing sublattice totally real number transmission system Z-basis