automorphism group (linear code)

Let $\\mathbb{F}_{q}$ be the finite field with $q$ 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 $\\mathbb{F}_{q}$ ofblock length $n$ 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

C_{M}:=\\left\\{\\left(\\sum\\limits_{i=1}^{n}M_{i1}c_{i},\\ldots,\\sum\\limits_{i=1}^%{n}M_{in}c_{i}\\right)\\mid(c_{1},\\ldots,c_{n})\\in C\\right\\}.

This definition looks quite complicated, but since $M$ is , itreally just means that $C_{M}$ is the linear code obtained from $C$ bypermuting its coordinates and then multiplying each coordinate withsome nonzero element from $\\mathbb{F}_{q}$ .

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

Sometimes one is only interested in the action of the permutationmatrices on $\\mathfrak{C}_{n,q}$ . The permutation matrices form a subgroupof $\\mathcal{M}_{n,q}$ and the resulting subgroup of the automorphism group $\\operatorname{Aut}(C)$ of a linear code $C\\in\\mathfrak{C}_{n,q}$ is called thepermutation group. In the case of binary codes, this doesn’tmake any difference, since the finite field $\\mathbb{F}_{2}$ contains onlyone 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