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**Lattices and codes. A course partially based on lectures by F. Hirzebruch.
2nd ed.**
*(English)*
Zbl 1030.11030

Advanced Lectures in Mathematics. Wiesbaden: Vieweg. xviii, 188 p. (2002).

The textbook under review is the second edition of course notes delivered by the author (Eindhoven 1988/89) partially based upon lectures from F. Hirzebruch (Bonn 1986/87). The first edition was reviewed in Zbl 0805.11048. Since that review was in German, let us reiterate the main topics. The focus of the book is on the relationship between algebraic coding theory and the theory of integral lattices. In particular, root lattices, the theory of modular forms, and even unimodular lattices, with special attention paid to the Leech lattice, are presented. Self-dual codes over \(F_p\) and their theta functions are also discussed in detail.

Besides correcting some errors contained in the first edition, the second contains additional exercises as well as a new section on Conway’s result about a relation between the Leech lattice and the automorphism group of the 26-dimensional unimodular hyperbolic lattice. Furthermore, additional basic material is given to make this text even more self-contained, rendering it suitable for advanced undergraduate students, graduate students, and anyone interested in the topics at hand.

Besides correcting some errors contained in the first edition, the second contains additional exercises as well as a new section on Conway’s result about a relation between the Leech lattice and the automorphism group of the 26-dimensional unimodular hyperbolic lattice. Furthermore, additional basic material is given to make this text even more self-contained, rendering it suitable for advanced undergraduate students, graduate students, and anyone interested in the topics at hand.

Reviewer: Katherine Roegner (Berlin)

### MSC:

11H06 | Lattices and convex bodies (number-theoretic aspects) |

94B05 | Linear codes (general theory) |

11F11 | Holomorphic modular forms of integral weight |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |

11F27 | Theta series; Weil representation; theta correspondences |

11E12 | Quadratic forms over global rings and fields |

11H71 | Relations with coding theory |