<think> Alright, let's tackle this problem step by step. So, we have a generator matrix G for a ternary linear code C in F_3^7. The order of the group of monomial automorphisms of C is given as 24. We need to find how many generator matrices produce codes that are monomially equivalent to C.

First, I remember that two codes are monomially equivalent if one can be obtained from the other by permuting coordinates and/or multiplying coordinates by non-zero scalars. In terms of generator matrices, this means that any such matrix can be transformed into another equivalent one via these operations.

The key here is understanding how many distinct generator matrices exist under these transformations. Since the automorphism group has order 24, each code C has exactly 24 monomially equivalent codes. However, we need to consider all possible generator matrices that could generate such codes.

But wait, I think there's a formula related to this. The number of generator matrices equivalent to G is equal to the product of the order of the automorphism group and the number of distinct generator matrices for each code in its equivalence class. However, since we're considering all possible monomially equivalent codes, it might just be that each code has 24 different generator matrices.

Hold on, maybe I'm overcomplicating this. Let me think again. The automorphism group's order is the number of ways to transform G into another generator matrix while keeping the code structure intact. So, if there are 24 such transformations, then for each code in the equivalence class, there are 24 different generator matrices.

But actually, no. Each code can be generated by multiple generator matrices, and the automorphism group's order tells us how many distinct generator matrices correspond to a single code. Wait, that might not be correct either.

I think I need to recall that for each code C, the number of monomially equivalent codes is equal to the size of the orbit under the action of the monomial group. The orbit-stabilizer theorem says that the size of the orbit times the size of the stabilizer (which is the automorphism group) equals the order of the group acting on it.

But in this case, we're dealing with generator matrices, not codes. So perhaps each code has multiple generator matrices, and the number of such matrices is related to the automorphism group's order.

Wait, let me clarify: The monomial automorphism group acts on the set of generator matrices by permuting columns and scaling them. Each orbit under this action corresponds to a monomial equivalence class of codes. So, if G is a generator matrix for C, then all matrices in its orbit are generator matrices for codes equivalent to C.

The size of the orbit would be equal to the order of the group divided by the stabilizer's order. But here, the automorphism group's order is given as 24, which I think is the stabilizer's order because it's the set of transformations that leave the code invariant.

So, if the monomial group acting on F_3^7 has a certain size, and the stabilizer (automorphism group) has order 24, then the orbit size would be |monomial group| / |stabilizer|. But I don't know the size of the entire monomial group.

Wait, perhaps that's not necessary. Maybe each code can have multiple generator matrices, and the number is equal to the order of the automorphism group multiplied by something else.

Alternatively, considering that for a given code C, any generator matrix G' equivalent to G must be related by a monomial transformation. The number of such transformations is exactly the size of the automorphism group, which is 24. Therefore, each code has 24 different generator matrices that are monomially equivalent.

But wait, no. Because if you have two different generator matrices for the same code, they might be related by an automorphism. So the number of distinct generator matrices for a single code would be equal to the size of the automorphism group divided by something? Or perhaps it's the other way around.

I'm getting confused here. Let me try another approach. The problem is asking how many generator matrices produce codes that are monomially equivalent to C. So, each such matrix corresponds to a code in the equivalence class of C under monomial transformations.

The number of such codes is equal to the size of the orbit of G under the action of the monomial group. But without knowing the size of the entire monomial group, it's hard to compute directly.

Wait, but perhaps we can use the fact that each code has a certain number of generator matrices, and the automorphism group's order is 24. So if there are N codes equivalent to C, then each contributes 24 generator matrices, so total would be N * 24.

But I don't know N either. Alternatively, maybe each code can have multiple generator matrices, but the number of distinct codes is equal to the orbit size, which is |monomial group| / |automorphism group|. But without knowing |monomial group|, this approach isn't helpful.

Wait a second, perhaps there's another way. The number of generator matrices equivalent to G is equal to the order of the monomial automorphism group multiplied by something else? Or maybe it's just 24 times the number of possible generator matrices for each code.

I'm stuck here. Let me try to look up some related concepts. I recall that in coding theory, two codes are monomially equivalent if there exists a monomial matrix (a product of permutation and diagonal matrices) that transforms one generator matrix into another. The set of all such transformations forms the automorphism group.

So, for each code C, the number of distinct generator matrices is equal to the size of its automorphism group. Wait, no, because different generator matrices can generate the same code. So perhaps the number of generator matrices equivalent to G is equal to the order of the automorphism group times something else.

Alternatively, maybe it's simply that each code has exactly 24 distinct generator matrices, so the total number would be 24 multiplied by the number of codes in its equivalence class.

But without knowing how many codes are monomially equivalent to C, I can't compute this directly. Maybe there's a formula or theorem that relates these quantities.

Wait, perhaps the key is that each code has exactly |Aut(C)| generator matrices up to monomial equivalence. But no, that doesn't sound right because Aut(C) counts the number of transformations preserving the code, not the number of distinct generators.

I think I need to approach this differently. Let's consider that for a given code C, any generator matrix G' can be obtained from G by permuting columns and scaling them. The set of all such G' forms an orbit under the action of the monomial group. The size of this orbit is equal to |monomial group| / |stabilizer|. Here, the stabilizer is Aut(C), which has order 24.

But I don't know the size of the entire monomial group. However, for F_3^7, the monomial group is the set of all monomial matrices, which are products of permutation matrices and diagonal matrices with non-zero entries. The number of such matrices is (number of permutations) * (number of possible diagonals). For F_3, each diagonal entry can be 1 or 2, so there are 2^7 possibilities for the diagonal. And the number of permutations is 7!.

So, |monomial group| = 7! * 2^7.

Calculating that: 7! is 5040, and 2^7 is 128. So 5040 * 128 = let's compute that.

5040 * 128: 5040 * 100 = 504,000; 5040 * 28 = 141,120. So total is 504,000 + 141,120 = 645,120.

So |monomial group| = 645,120.

Now, the orbit size is |monomial group| / |Aut(C)| = 645,120 / 24 = let's compute that.

645,120 divided by 24: 24 * 27,000 = 648,000. So 645,120 - 648,000 is negative, so maybe I miscalculated.

Wait, let me compute 645,120 / 24:

Divide numerator and denominator by 24: 24 * 27,000 = 648,000. But 645,120 is less than that. So perhaps I should do it step by step.

Compute 645,120 ÷ 24:

24 * 26,000 = 624,000

Subtract: 645,120 - 624,000 = 21,120

Now, 24 * 880 = 21,120 (since 24900=21,600 which is too much; 24880=21,120)

So total is 26,000 + 880 = 26,880.

Therefore, the orbit size is 26,880. This means there are 26,880 distinct generator matrices that produce codes monomially equivalent to C.

Wait, but does this make sense? Because each code in the equivalence class can have multiple generator matrices, and the total number of such matrices would be the orbit size times something else?

No, actually, no. The orbit size is the number of distinct generator matrices under the action of the monomial group. Each orbit corresponds to a unique code up to monomial equivalence. So if G is in an orbit of size 26,880, then there are 26,880 different generator matrices that produce codes equivalent to C.

But wait, no. Because each code can have multiple generator matrices. For example, two different generator matrices might generate the same code but be related by a monomial transformation. So in this case, the orbit size would actually correspond to the number of distinct generator matrices for all codes equivalent to C.

Wait, perhaps I'm conflating codes and their generator matrices. Let me clarify:

• Each code can have multiple generator matrices.
• Two generator matrices are equivalent if they generate the same code up to monomial transformations.
• The orbit of a generator matrix G under the monomial group action consists of all generator matrices that produce codes equivalent to C.

Therefore, the size of this orbit is 26,880. So there are 26,880 different generator matrices that produce codes monomially equivalent to C.

But wait, another thought: The number of distinct codes equivalent to C would be equal to the orbit size divided by the number of generator matrices per code. But I'm not sure about this.

Alternatively, perhaps each code in the equivalence class has exactly |Aut(C)| generator matrices. So if there are N codes in the equivalence class, then total generator matrices would be N * |Aut(C)|.

But from earlier, we have orbit size = 26,880, which is equal to |monomial group| / |Aut(C)|. So:

Orbit size = |M| / |Aut(C)| => 26,880 = 645,120 / 24.

So the number of distinct codes in the equivalence class would be orbit size divided by something else? Or perhaps each code has exactly |Aut(C)| generator matrices. So if there are N codes, then total generator matrices is N * |Aut(C)|.

But we have orbit size = 26,880, which counts all possible generator matrices for all equivalent codes. Therefore, the number of distinct codes would be 26,880 / |Aut(C)|, because each code has |Aut(C)| generator matrices.

So N = 26,880 / 24 = 1,120.

Therefore, there are 1,120 distinct codes equivalent to C, and each has 24 generator matrices. So the total number of generator matrices is 1,120 * 24 = 26,880, which matches our orbit size.

But wait, that seems circular because we started with orbit size being 26,880. So perhaps I'm overcomplicating it.

In any case, the problem asks for how many generator matrices produce codes monomially equivalent to C. That would be the orbit size of G under the monomial group action, which is |M| / |Aut(C)| = 26,880.

Therefore, the answer should be 26,880. </think>

[THOUGHT] To determine how many generator matrices produce codes monomially equivalent to ( \mathcal{C} ), we consider the action of the monomial group on the set of generator matrices. The number of such matrices is given by the orbit-stabilizer theorem: the size of the orbit (number of equivalent generator matrices) equals the order of the monomial group divided by the order of the stabilizer (automorphism group).

The monomial group for ( \mathbb{F}_3^7 ) consists of all products of permutation and diagonal matrices. Its order is calculated as ( 7! \times 2^7 = 5040 \times 128 = 645,120 ). Given the automorphism group's order is 24, the number of equivalent generator matrices is ( 645,120 / 24 = 26,880 ). [/THOUGHT]

[PYTHON]

Calculate the size of the monomial group

monomial_group_size = (7 * 6 * 5 * 4 * 3 * 2 * 1) * (2 ** 7) automorphism_order = 24

Number of equivalent generator matrices

equivalent_generator_matrices = monomial_group_size // automorphism_order print(equivalent_generator_matrices)

<think> Okay, so I have this problem about generator matrices for a ternary linear code. The generator matrix G is given, and it's part of a code C in F_3^7. The question is asking how many generator matrices produce codes that are monomially equivalent to C. It also mentions that the order of the group of monomial automorphisms of C is 24.

First, I need to understand what monomial equivalence means. From what I remember, two codes are monomially equivalent if one can be obtained from the other by permuting coordinates and scaling coordinates by non-zero scalars. So, in this case, we're looking at all possible generator matrices that can be transformed into G through these operations.

The key here is to use group theory, specifically the orbit-stabilizer theorem. The number of monomially equivalent generator matrices should correspond to the size of the orbit of G under the action of the monomial group. According to the orbit-stabilizer theorem, the size of the orbit is equal to the order of the group divided by the order of the stabilizer (which is given as 24).

So, I need to find the order of the monomial group acting on the generator matrices. The monomial group for F_3^7 consists of all invertible monomial matrices. An invertible monomial matrix is a product of a permutation matrix and a diagonal matrix with non-zero entries.

The number of such matrices can be calculated as follows:

• There are 7! ways to permute the coordinates.
• For each coordinate, there are 2 choices for scaling (since F_3 has elements {1, 2}, excluding 0). So, for 7 coordinates, that's 2^7 possibilities.

Therefore, the order of the monomial group is 7! * 2^7. Let me compute that: 7! = 5040 2^7 = 128 So, 5040 * 128 = 645,120

Now, using the orbit-stabilizer theorem, the number of distinct generator matrices is the order of the monomial group divided by the order of the stabilizer (which is given as 24). So:

Number of equivalent generator matrices = 645,120 / 24 = 26,880

Wait, but I should double-check this calculation to make sure there are no errors. Let me verify:

645,120 divided by 24: First, divide both numerator and denominator by 24: 645,120 / 24 = (645,120 ÷ 24) = 26,880

Yes, that seems correct. So the number of generator matrices producing codes monomially equivalent to C is 26,880.

I think this makes sense because each equivalence class under monomial transformations has size equal to the group order divided by the stabilizer's order. Since the automorphism group's order is given as 24, it fits into the formula correctly. </think>

The number of generator matrices that produce codes monomially equivalent to ( \mathcal{C} ) is calculated using the orbit-stabilizer theorem. The size of the monomial group is divided by the order of the stabilizer (automorphism group), resulting in 26,880.

\boxed{26880}