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 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 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 equivalent. The isotropy subgroup of $C$ is its automorphism group, denoted by $\Aut(C)$ . The elements of $\Aut(C)$ are the 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 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.