# 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 $\mathbb{F}_{q}$ of block 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 , 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 $\mathbb{F}_{q}$.

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

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 $\operatorname{Aut}(C)$ 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.

 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