## Abstract

Revisiting an approach by Conway and Sloane we investigate a collection of optimal non-linear binary codes and represent them as (non-linear) codes over Z(4). The Fourier transform will be used in order to analyze these codes, which leads to a new algebraic representation involving subgroups of the group of units in a certain ring.

One of our results is a new representation of Best's (10, 40, 4) code as a coset of a subgroup in the group of invertible elements of the group ring Z(4)[Z(5)]. This yields a particularly simple algebraic decoding algorithm for this code.

The technique at hand is further applied to analyze Julin's (12, 144, 4) code and the (12, 24, 12) Hadamard code. It can also be used in order to construct a (non-optimal) binary (14, 56, 6) code.

Original language | English |
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Title of host publication | 2012 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY PROCEEDINGS (ISIT) |

Publisher | IEEE |

Pages | 3115-3119 |

Number of pages | 5 |

ISBN (Electronic) | 978-1-4673-2579-0 |

ISBN (Print) | 978-1-4673-2580-6 |

DOIs | |

Publication status | Published - 2012 |

MoE publication type | A4 Article in a conference publication |

Event | IEEE International Symposium on Information Theory - Cambridge, Morocco Duration: 1 Jul 2012 → 6 Jul 2012 |

### Publication series

Name | IEEE International Symposium on Information Theory |
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Publisher | IEEE |

Name | Information Theory Proceedings (ISIT) |
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Publisher | IEEE |

ISSN (Print) | 2157-8095 |

ISSN (Electronic) | 2157-8117 |

### Conference

Conference | IEEE International Symposium on Information Theory |
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Country | Morocco |

City | Cambridge |

Period | 01/07/2012 → 06/07/2012 |