PlanetMath: automorphism group (linear code)

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

automorphism group (linear code)

(Definition)

Let $\mathbb{F}_q$ be the finite field with elements. The group $\mathcal {M}_{n,q}$ of $n\times n$ monomial matrices with entries in $\mathbb{F}_q$ acts on the set $\mathfrak{C}_{n,q}$ of linear codes over $\mathbb{F}_q$ of block length via the monomial transform: let $M=(M_{ij})_{i,j=1}^n\in\mathcal {M}_{n,q}$ and $C\in\mathfrak{C}_{n,q}$ and set

$\displaystyle C_M:=\left\{\left(\sum\limits _{i=1}^nM_{i1}c_i,\ldots,\sum\limits _{i=1}^nM_{in}c_i\right)\mid(c_1,\ldots,c_n)\in C\right\}.$

This definition looks quite complicated, but since

is monomial, it really just means that

is the linear code obtained from

by permuting its coordinates and then multiplying each coordinate with some nonzero element from $\mathbb{F}_q$ .

Two linear codes lying in the same orbit with respect to this action are said to be equivalent. The isotropy subgroup of is its automorphism group, denoted by ${\mathrm{Aut}}(C)$ . The elements of ${\mathrm{Aut}}(C)$ are the automorphisms of .

Sometimes one is only interested in the action of the permutation matrices on $\mathfrak{C}_{n,q}$ . The permutation matrices form a subgroup of $\mathcal {M}_{n,q}$ and the resulting subgroup of the automorphism group ${\mathrm{Aut}}(C)$ of a linear code $C\in\mathfrak{C}_{n,q}$ is called the permutation group. In the case of binary codes, this doesn't make any difference, since the finite field $\mathbb{F}_2$ contains only one nonzero element.

"automorphism group (linear code)" is owned by GrafZahl.

(view preamble)