## Designs and Their CodesAlgebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. |

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### Contents

Designs | 1 |

Codes | 25 |

The geometry of vector spaces | 89 |

Symmetric Designs | 117 |

The standard geometric codes | 139 |

Codes from planes | 199 |

Hadamard designs | 249 |

Steiner systems | 295 |

Bibliography | 317 |

Glossary | 337 |

344 | |

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### Common terms and phrases

2-design abelian affine geometry affine plane algebra Assmus automorphism group Baer subplane binary code bound CF(S Chapter characteristic functions clearly codeword coding theory columns construction contains coordinate Corollary corresponding cosets cyclic code defined Definition denote desarguesian design of points difference set dimension divides element entries equation equivalence class example fact Fano plane finite follows give given Hadamard matrices Hamming code hence incidence matrix incidence structure incidence vectors integer involution isomorphic Lemma linear code minimum weight minimum-weight vectors modulo Moreover non-zero notation number of blocks number of points obtained orthogonal oval parameters parity-check matrix permutation PG(V plane of order point set polynomial precisely prime projective geometry projective plane projective space Proof Proposition r-flats Reed-Muller codes result scalar multiples Section self-dual subcode subgroup subset subspace symmetric designs tangent Theorem transform translation plane unique vector space vectors of weight weight enumerator