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Group Superset Group Action Frame Subgroup Finite Field Stabilizer Orbit Code Linear Code Permutation Matrix Monomial Matrix

1	Frame
1	Superset
1	Group Action
1	Group
1	Subgroup
3	Finite Field
4	Code
5	Stabilizer
5	Orbit
6	Permutation Matrix
7	Linear Code
11	Monomial Matrix

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Automorphism Group Linear Code

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.