I am trying to show that the linear transformation of two monomially-equivalent linear codes preserves the minimum distance and the two equivalent codes have the same dimension.
First, what is the definition of monomially-equivalent codes: So, we say that the codes $C$ and $\tilde{C}$ are monomially-equivalent, if there exist some non-zero scalars $\gamma _{1},...\gamma _{n}$ and a permutation $\sigma $ of $\left \{ 1,...,n \right \}$ such that $\left ( c_{1},...,c_{n} \right )\in C\Leftrightarrow \left ( \gamma _{1}c_{\sigma (1)},...,\gamma _{n}c_{\sigma (n)} \right )\in \tilde{C}$.
The linear transformation i tried to define as $T:G\rightarrow P\tilde{G}M$, where $G$ and $\tilde{G}$ are the generator matrices of $C$ and $\tilde{C}$, $P$ is an invertible matrix and $M$ is matrix, which has some unique $\gamma _{i}$ as the only element in each column, every other positions are $0$.
Do i have to show that $T$ must be an isometry, since it the linear transformation should preserve the minimum distance? How to show that the codes have the same dimension? Can anybody help me with those questions, please?
Thank you in advance!
Consider the map $P_\sigma \Gamma : C \to \tilde{C}$ where $\Gamma = \operatorname{diag}(\gamma_1, \ldots, \gamma_n)$ and $P_\sigma$ is the permutation matrix corresponding to $\sigma$. Note that $\tilde{C} = C P_\sigma \Gamma (= \{ x P_\sigma \Gamma \mid x \in C \})$.
The matrix $\Gamma$ is full rank since $\gamma_1, \cdots, \gamma_n$ are all nonzero. Any permutation matrix is full rank, so $P_\sigma$ in particular is. Hence $P_\sigma \Gamma$ is full rank and so $\dim(C) = \dim(\tilde{C})$.
As $\gamma_1, \cdots, \gamma_n$ are all nonzero,
$$ w((c_1, \ldots, c_n) P_\sigma \Gamma) = w(\gamma_1 c_{\sigma(1)}, \ldots, \gamma_n c_{\sigma(n)}) = w(c_1, \ldots, c_n). $$
That is, $P_\sigma \Gamma$ is a linear isometry with respect to the Hamming weight. Therefore
$$ d_\text{min}(C) = \min_{\substack{x,y \in C\\ x \ne y}} w(x-y) = \min_{\substack{x,y \in C\\ x \ne y}} w((x-y) P_\sigma \Gamma) = \min_{\substack{x,y \in \tilde{C}\\ x \ne y}} w(x-y) = d_\text{min}(\tilde{C}), $$
noting that $d(x,y) = w(x-y)$.