# dual code

Let $C$ be a linear code of block length $n$ over the finite field^{}
${\mathrm{\pi \x9d\x94\xbd}}_{q}$. Then the set

$${C}^{\beta \x9f\x82}:=\{d\beta \x88\x88{\mathrm{\pi \x9d\x94\xbd}}_{q}^{n}\beta \x88\pounds c\beta \x8b\x85d=0\beta \x81\u2019\text{\Beta for all\Beta}\beta \x81\u2019c\beta \x88\x88C\}$$ |

is the *dual code* of $C$. Here, $c\beta \x8b\x85d$ denotes either the
standard dot product^{} or the Hermitian dot product.

This definition is reminiscent of orthogonal complements^{} of http://planetmath.org/node/5398finite
dimensional vector spaces^{} over the real or complex numbers^{}. Indeed,
${C}^{\beta \x9f\x82}$ is also a linear code and it is true that if $k$ is the
http://planetmath.org/node/5398dimension^{} of $C$, then the of
${C}^{\beta \x9f\x82}$ is $n-k$. It is, however, not necessarily true that
$C\beta \x88\copyright {C}^{\beta \x9f\x82}=\{0\}$. For example, if $C$ is the binary code of block
length $2$ http://planetmath.org/node/806spanned by the codeword $(1,1)$ then $(1,1)\beta \x8b\x85(1,1)=0$,
that is, $(1,1)\beta \x88\x88{C}^{\beta \x9f\x82}$. In fact, $C$ equals ${C}^{\beta \x9f\x82}$ in this
case. In general, if $C={C}^{\beta \x9f\x82}$, $C$ is called
*self-dual*. Furthermore $C$ is called *self-orthogonal* if
$C\beta \x8a\x86{C}^{\beta \x9f\x82}$.

Famous examples of self-dual codes are the extended binary Hamming code of block length $8$ and the extended binary Golay code of block length $24$.

Title | dual code |
---|---|

Canonical name | DualCode |

Date of creation | 2013-03-22 15:13:29 |

Last modified on | 2013-03-22 15:13:29 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 6 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 94B05 |

Related topic | LinearCode |

Related topic | OrthogonalComplement |

Defines | self-dual |

Defines | self-orthogonal |