Title | **Gr\"obner bases and combinatorics for binary codes.** |

Author(s) | Mijail Borges-Quintana, Miguel A. Borges-Trenard, Patrick Fitzpatrick, Edgar Martinez-Moro |

Type | Article in Journal |

Abstract | In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. In the second one, by associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis. |

Keywords | Binary codes, Cycle bases, Gröbner bases |

ISSN | 0938-1279; 1432-0622/e |

URL |
http://link.springer.com/article/10.1007%2Fs00200-008-0080-2 |

Language | English |

Journal | Appl. Algebra Eng. Commun. Comput. |

Volume | 19 |

Number | 5 |

Pages | 393--411 |

Publisher | Springer, Berlin/Heidelberg |

Year | 2008 |

Edition | 0 |

Translation |
No |

Refereed |
No |