Designs and their codes.

*(English)*Zbl 0762.05001
Cambridge Tracts in Mathematics. 103. Cambridge: Cambridge University Press. x, 352 p. (1992).

The aim of this book, as accurately defined by the authors, is to study applications of algebraic coding theory to the analysis and classification of combinatorial designs. Much of the content has been used as a basis for courses at both Clemson and Lehigh Universities. The exposition is a fine blend of material that is easily accessible to the student with more advanced topics that will be of great interest for the researcher.

The book consists of preface, 8 chapters, bibliography of 301 titles, glossary, and indices of names and terms. The glossary and the index of terms can be used as a dictionary that facilitates the more advanced reader to start directly from the chapter of any particular interest. The first five chapters are of a background nature. Chapter 1 is a short introduction to combinatorial designs with an emphasis to the rank of the incidence matrix. Chapter 2 introduces linear codes and cyclic, Hamming, MDS and quadratic-residues codes. Chapter 3 is a short course in finite geometry. Chapter 4 discusses symmetric designs and an interesting relation between the code of a design and the multiplier theorem. Chapter 5 contains a systematic presentation of the standard geometric codes: Reed-Muller codes and their generalizations.

The last three chapters are devoted to methods for analysis of designs by means of the codes of their incidence matrices. Chapter 6 contains an original study of the codes of finite planes and some remarkable relations between the code of an affine plane and the code of the corresponding projective plane, as well as geometric characterizations of codewords as lines, unitals, or ovals. Chapter 7 is a study of Hadamard matrices from the point of view of coding theory. The last Chapter 8 discusses some classes of Steiner designs that are amenable to a treatment via coding theory: the small and large Witt designs, Steiner triple and quadruple systems, unitals and oval designs, as well as some other classes of 3-designs.

The book is very appropriate as a text in various courses in discrete mathematics, combinatorics, coding theory, finite geometry, and will invaluable for researchers in these fields.

The book consists of preface, 8 chapters, bibliography of 301 titles, glossary, and indices of names and terms. The glossary and the index of terms can be used as a dictionary that facilitates the more advanced reader to start directly from the chapter of any particular interest. The first five chapters are of a background nature. Chapter 1 is a short introduction to combinatorial designs with an emphasis to the rank of the incidence matrix. Chapter 2 introduces linear codes and cyclic, Hamming, MDS and quadratic-residues codes. Chapter 3 is a short course in finite geometry. Chapter 4 discusses symmetric designs and an interesting relation between the code of a design and the multiplier theorem. Chapter 5 contains a systematic presentation of the standard geometric codes: Reed-Muller codes and their generalizations.

The last three chapters are devoted to methods for analysis of designs by means of the codes of their incidence matrices. Chapter 6 contains an original study of the codes of finite planes and some remarkable relations between the code of an affine plane and the code of the corresponding projective plane, as well as geometric characterizations of codewords as lines, unitals, or ovals. Chapter 7 is a study of Hadamard matrices from the point of view of coding theory. The last Chapter 8 discusses some classes of Steiner designs that are amenable to a treatment via coding theory: the small and large Witt designs, Steiner triple and quadruple systems, unitals and oval designs, as well as some other classes of 3-designs.

The book is very appropriate as a text in various courses in discrete mathematics, combinatorics, coding theory, finite geometry, and will invaluable for researchers in these fields.

Reviewer: V.D.Tonchev (Houghton)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

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

51Exx | Finite geometry and special incidence structures |

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

05Bxx | Designs and configurations |