# perfect code

Let $C$ be a linear (http://planetmath.org/LinearCode) $(n,k,d)$-code over ${\mathbb{F}}_{q}$.

The *packing radius* of $C$ is defined to be the value

$\rho (C)={\displaystyle \frac{d-1}{2}}.$ |

The *covering radius* of $C$ is

$r(C)=\underset{x}{\mathrm{max}}\underset{c}{\mathrm{min}}\delta (x,c)$ |

with $x\in {\mathbb{F}}_{q}^{n}$ and $c\in C$, and where $\delta $ denotes the Hamming distance^{} on ${\mathbb{F}}_{q}^{n}$.

The code (http://planetmath.org/Code) $C$ is said to be *perfect* if $r(C)=\rho (C)$.

The list of of linear perfect codes is very short, including only trivial codes, Hamming codes (i.e. $\rho =1$), and the binary and ternary Golay (http://planetmath.org/BinaryGolayCode) codes.

Title | perfect code |
---|---|

Canonical name | PerfectCode |

Date of creation | 2013-03-22 14:23:43 |

Last modified on | 2013-03-22 14:23:43 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11T71 |

Defines | packing radius |

Defines | covering radius |