# automorphism group (linear code)

Let ${\mathrm{\pi \x9d\x94\xbd}}_{q}$ be the finite field with $q$ elements. The group
${\mathrm{\beta \x84\xb3}}_{n,q}$ of $n\Gamma \x97n$ monomial matrices with entries in ${\mathrm{\pi \x9d\x94\xbd}}_{q}$
acts on the set ${\mathrm{\beta \x84\xad}}_{n,q}$ of linear codes^{} over ${\mathrm{\pi \x9d\x94\xbd}}_{q}$ of
block length $n$ via the *monomial transform*: let $M={({M}_{i\beta \x81\u2019j})}_{i,j=1}^{n}\beta \x88\x88{\mathrm{\beta \x84\xb3}}_{n,q}$ and $C\beta \x88\x88{\mathrm{\beta \x84\xad}}_{n,q}$ and set

$${C}_{M}:=\{(\underset{i=1}{\overset{n}{\beta \x88\x91}}{M}_{i\beta \x81\u20191}\beta \x81\u2019{c}_{i},\mathrm{\beta \x80\xa6},\underset{i=1}{\overset{n}{\beta \x88\x91}}{M}_{i\beta \x81\u2019n}\beta \x81\u2019{c}_{i})\beta \x88\pounds ({c}_{1},\mathrm{\beta \x80\xa6},{c}_{n})\beta \x88\x88C\}.$$ |

This definition looks quite complicated, but since $M$ is , it
really just means that ${C}_{M}$ is the linear code obtained from $C$ by
permuting its coordinates^{} and then multiplying each coordinate with
some nonzero element from ${\mathrm{\pi \x9d\x94\xbd}}_{q}$.

Two linear codes lying in the same orbit with respect to this action
are said to be *equivalent ^{}*. The isotropy subgroup

^{}of $C$ is its

*automorphism group*, denoted by $\mathrm{Aut}\beta \x81\u2018(C)$. The elements of $\mathrm{Aut}\beta \x81\u2018(C)$ are the

^{}*automorphisms*of $C$.

^{}Sometimes one is only interested in the action of the permutation
matrices^{} on ${\mathrm{\beta \x84\xad}}_{n,q}$. The permutation matrices form a subgroup^{}
of ${\mathrm{\beta \x84\xb3}}_{n,q}$ and the resulting subgroup of the automorphism group
$\mathrm{Aut}\beta \x81\u2018(C)$ of a linear code $C\beta \x88\x88{\mathrm{\beta \x84\xad}}_{n,q}$ is called the
*permutation group ^{}*. In the case of binary codes, this doesnβt
make any difference

^{}, since the finite field ${\mathrm{\pi \x9d\x94\xbd}}_{2}$ contains only one nonzero element.

Title | automorphism group (linear code) |

Canonical name | AutomorphismGrouplinearCode |

Date of creation | 2013-03-22 15:18:40 |

Last modified on | 2013-03-22 15:18:40 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 5 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 94B05 |

Synonym | automorphism group |

Related topic | LinearCode |

Defines | monomial transform |

Defines | equivalent |

Defines | equivalent code |

Defines | automorphism |

Defines | permutation group |