Let
be a generator matrix of a ternary linear code . It is known that the order of the group of monomial automorphisms of is 24. How many generator matrices produce codes that are monomially equivalent to ?\
To determine how many generator matrices produce codes that are monomially equivalent to (\mathcal{C}), we follow these steps:
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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.
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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.
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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.
[ \boxed{24} ]