I've been reading about equivalent codes, and the topic of monomial automorphisms came up. These are the set of monomial matrices (square matrices with exactly one nonzero entry in each row and column) that map a code $\mathcal{C}$ to itself. There are several properties mentioned that I am having trouble verifying, and I was wondering if anyone can help. The properties are:
For diagonal matrices $D_i$, permutation matrices $P_i$ and automorphisms $\gamma_i$ of the field $\mathbb{F}_q$:
(a) $(D_1P_1\gamma_1)(D_2P_2\gamma_2) = (D_3P_3\gamma_3)$
(b) $(D_4P_4\gamma_4)^{-1} = D_5P_5\gamma_5$
For (a), I am having trouble finding $D_3$, $P_3$, and $\gamma_3$ in terms of $D_1, D_2, P_1, P_2, \gamma_1,$ and $\gamma_2$. For (b), I am having trouble finding $D_5, P_5,$ and $\gamma_5$ in terms of $D_4, P_4,$ and $\gamma_4$.
Any help would be greatly appreciated.
EDIT
If $\gamma$ is a field automorphism of $\mathbb{F}_q$, $\sigma$ is the permutation associated with $P$, and $M = DP$ is a monomial map with entries in $\mathbb{F}_q$, then applying the map $M\gamma$ to a vector $x$ goes as follows:
- Multiply the $i$th component of $x$ by the $i$th diagonal entry of $D$,
- Move this product to the coordinate position $i\sigma$,
- Apply $\gamma$ to this component.
So basically, you apply the permutaion to the diagonal matrix, and then you apply the field automorphism on the right of the elements in the resulting matrix.