automorphism group (linear code)AutomorphismGroupLinearCode2005-05-25 12:58:062005-05-25 13:02:01Definitionautomorphismmonomial transformequivalentequivalent codeautomorphismpermutation groupcodetransform% this is the default PlanetMath preamble. as your knowledge
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Let $\mbb{F}_q$ be the finite field with $q$ elements. The group
$\mc{M}_{n,q}$ of $n\times n$ monomial matrices with entries in $\mbb{F}_q$
acts on the set $\mf{C}_{n,q}$ of linear codes over $\mbb{F}_q$ of
block length $n$ via the \emph{monomial transform}: let $M=(M_{ij})_{i,j=1}^n\in\mc{M}_{n,q}$ and $C\in\mf{C}_{n,q}$ and set
\begin{equation*}
C_M:=\left\{\left(\Sum_{i=1}^nM_{i1}c_i,\ldots,\Sum_{i=1}^nM_{in}c_i\right)\mid(c_1,\ldots,c_n)\in C\right\}.
\end{equation*}
This definition looks quite complicated, but since $M$ is \PMlinkescapetext{monomial}, 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 $\mbb{F}_q$.
Two linear codes lying in the same orbit with respect to this action
are said to be \emph{equivalent}. The isotropy subgroup of $C$
is its \emph{automorphism group}, denoted by $\Aut(C)$. The elements
of $\Aut(C)$ are the \emph{automorphisms} of $C$.
Sometimes one is only interested in the action of the permutation
matrices on $\mf{C}_{n,q}$. The permutation matrices form a subgroup
of $\mc{M}_{n,q}$ and the resulting subgroup of the automorphism group
$\Aut(C)$ of a linear code $C\in\mf{C}_{n,q}$ is called the
\emph{permutation group}. In the case of binary codes, this doesn't
make any difference, since the finite field $\mbb{F}_2$ contains only
one nonzero element.