# linear code

Often in coding , a code’s alphabet is taken to be a finite field. In particular, if $A$ is the finite field with two (resp. three, four, etc.) elements, we call $C$ a binary (resp. ternary, quaternary, etc.) code. In particular, when our alphabet is a finite field then the set ${A}^{n}$ is a vector space^{} over $A$, and we define a *linear code ^{} over $A$* of block length $n$ to be a subspace

^{}(as opposed to merely a subset) of ${A}^{n}$. We define the

*dimension*to be its dimension as a vector space over $A$.

^{}of $C$Though not sufficient for unique classification, a linear code’s block length, dimension, and minimum distance are three crucial parameters in determining the strength of the code. For referencing, a linear code with block length $n$, dimension $k$, and minimum distance $d$ is referred to as an $(n,k,d)$-code.

Some examples of linear codes are Hamming Codes, BCH codes, Goppa codes, Reed-Solomon codes, and the Golay code (http://planetmath.org/BinaryGolayCode).

Title | linear code |

Canonical name | LinearCode |

Date of creation | 2013-03-22 14:21:24 |

Last modified on | 2013-03-22 14:21:24 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 94B05 |

Related topic | CyclicCode |

Related topic | WeightEnumerator |

Related topic | DualCode |

Related topic | EvenCode |

Related topic | AutomorphismGroupLinearCode |

Defines | binary code |

Defines | ternary code |

Defines | quaternary code |

Defines | dimension of a linear code |