Codes and designs.

*(English)*Zbl 0922.94011
Pless, V. S. (ed.) et al., Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 1229-1267 (1998).

{For the entire collection see Zbl 0907.94001 }.

The nine sections of this handbook chapter survey the most important connections between codes and combinatorial designs. After the introduction, in Section 2, it is shown how binary codes achieving the Plotkin bound can be constructed from Hadamard matrices and designs. In Section 3, the correspondence between constant weight codes achieving the restricted and unrestricted Johnson bound and, respectively, 2-designs and Steiner systems is discussed. Equidistant \(q\)-ary codes are considered in Section 4. This topic is close to that of Section 2. The related designs are resolvable designs and orthogonal arrays. Only equidistant \(q\)-ary codes of a size that is a multiple of \(q\) are treated. Results due to Assmus and Mattson that show how \(t\)-designs can be constructed from perfect codes and linear codes are treated in Sections 5 and 6. Several generalizations of the Assmus-Mattson theorem are also stated. In Section 7, some constructions of self-dual codes from designs and Hadamard matrices are presented. In Section 8, it is shown how designs supported by a linear code can be used for decoding of the dual code. An interesting problem is therefore to find designs of minimum rank among designs with given parameters; many results on this problem are mentioned. Finally, in Section 9, methods from the theory of linear codes are utilized for construction and classification of certain types of designs such as quasi-symmetric designs.

The nine sections of this handbook chapter survey the most important connections between codes and combinatorial designs. After the introduction, in Section 2, it is shown how binary codes achieving the Plotkin bound can be constructed from Hadamard matrices and designs. In Section 3, the correspondence between constant weight codes achieving the restricted and unrestricted Johnson bound and, respectively, 2-designs and Steiner systems is discussed. Equidistant \(q\)-ary codes are considered in Section 4. This topic is close to that of Section 2. The related designs are resolvable designs and orthogonal arrays. Only equidistant \(q\)-ary codes of a size that is a multiple of \(q\) are treated. Results due to Assmus and Mattson that show how \(t\)-designs can be constructed from perfect codes and linear codes are treated in Sections 5 and 6. Several generalizations of the Assmus-Mattson theorem are also stated. In Section 7, some constructions of self-dual codes from designs and Hadamard matrices are presented. In Section 8, it is shown how designs supported by a linear code can be used for decoding of the dual code. An interesting problem is therefore to find designs of minimum rank among designs with given parameters; many results on this problem are mentioned. Finally, in Section 9, methods from the theory of linear codes are utilized for construction and classification of certain types of designs such as quasi-symmetric designs.

Reviewer: Patric Östergård (Helsinki)

##### MSC:

94B05 | Linear codes, general |

05B05 | Combinatorial aspects of block designs |

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

94B30 | Majority codes |

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

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

94B25 | Combinatorial codes |

94B65 | Bounds on codes |

05B30 | Other designs, configurations |