Algebraic number theory and code design for Rayleigh fading channels.

*(English)*Zbl 1137.94014
Found. Trends Commun. Inf. Theory 1, No. 3, 94 p. (2004). Foundations and Trends in Communications and Information Theory. Hanover, MA: NOW (ISBN 978-1-933019-07-9/pbk). vi, 88 p. (2004).

In recent years wireless communication requires code designs for fading channels. This book gives an overview on the use of lattices specially constructed for that purpose. It includes a tutorial on the prerequisites needed from algebraic number theory.

Following the introduction the authors explain models for fading channels in chapter 2. The next chapter describes basics of lattices. Chapter 4 deals with decoding methods. Decoding can be done by calculating closest lattice points. Though that problem is in general NP-hard the authors claim that the lattices which need to be considered for the appropriate codes behave much better in practice. This claim is supported by the results of numerous calculations. Chapter 5 contains fundamentals from algebraic number theory. Here, the authors also introduce the existing software KASH/KANT which they use for their computations. In chapters 6, 7 more advanced methods are described which were developed recently. These are lattices coming from ideals of algebraic number fields and so-called rotated lattice codes. The last chapter is on applications and conclusions. It also includes codes for MIMO channels (multiple input multiple output).

The book is pleasant to read. The authors include many illustrating examples and graphics. They also demonstrate the convenient use of existing software by presenting several short and easily written programs in the shell language KASH.

Following the introduction the authors explain models for fading channels in chapter 2. The next chapter describes basics of lattices. Chapter 4 deals with decoding methods. Decoding can be done by calculating closest lattice points. Though that problem is in general NP-hard the authors claim that the lattices which need to be considered for the appropriate codes behave much better in practice. This claim is supported by the results of numerous calculations. Chapter 5 contains fundamentals from algebraic number theory. Here, the authors also introduce the existing software KASH/KANT which they use for their computations. In chapters 6, 7 more advanced methods are described which were developed recently. These are lattices coming from ideals of algebraic number fields and so-called rotated lattice codes. The last chapter is on applications and conclusions. It also includes codes for MIMO channels (multiple input multiple output).

The book is pleasant to read. The authors include many illustrating examples and graphics. They also demonstrate the convenient use of existing software by presenting several short and easily written programs in the shell language KASH.

Reviewer: Michael Pohst (Berlin)

##### MSC:

94B99 | Theory of error-correcting codes and error-detecting codes |

94A40 | Channel models (including quantum) in information and communication theory |

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

11R99 | Algebraic number theory: global fields |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |