Problem Analysis #7

To determine how many generator matrices produce codes that are monomially equivalent to $\mathcal{C}$, we follow these steps:

1. Understand Monomial Equivalence: Two generator matrices $G$ and $H$ represent the same code under monomial equivalence if there exists a monomial matrix $M \in \text{GL}(4,3)$ such that $HM = G$. The group of monomial automorphisms has order 24.

2. Count Equivalent Generator Matrices: For each generator matrix equivalent to $G$, we can obtain it by applying a unique element from the automorphism group (up to scaling rows, which doesn't change the code). However, since row permutations and non-zero scalar multiplications are allowed within monomial matrices, but these operations preserve equivalence classes.

3. Calculate Total Equivalent Matrices: The number of distinct generator matrices equivalent to $G$ is equal to the order of the automorphism group (24), as each element corresponds to a unique transformation preserving code equivalence under scaling and permutation constraints in $\mathbb{F}_3^7$.

Thus, there are 24 such generator matrices.