[THOUGHT] We need to interpret the problem. We have a ternary linear code C with generator matrix G (4×7 over F_3). The order of the monomial automorphism group of C is given as 24. We are asked: "How many generator matrices produce codes that are monomially equivalent to C?" Possibly meaning: Given the code C, how many different generator matrices (4×7 over F_3) generate a code that is monomially equivalent to C? But that seems too broad because any invertible linear combination of rows yields the same code, and any monomial equivalence yields an equivalent code. Possibly the question is: Among all 4×7 matrices over F_3 of rank 4, how many generate a code monomially equivalent to C? But that's huge.

Maybe the intended meaning: Given the specific code C, we consider all generator matrices (i.e., all 4×7 matrices whose row space is C). How many such matrices are there? But that's just the number of ordered bases of C (since a generator matrix is a set of 4 linearly independent rows that span C). Over F_3, the number of ordered bases of a 4-dimensional subspace of F_3^7 is: (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = (81-1)(81-3)(81-9)(81-27) = 807872*54. But that's for any 4-dim subspace. However, the code C is fixed, so the number of generator matrices for C is exactly that. But the problem gives extra info: order of monomial automorphism group is 24. That suggests the answer might involve that number.

Maybe the question is: How many generator matrices produce codes that are monomially equivalent to C? Possibly meaning: Consider the set of all 4×7 matrices over F_3 of rank 4. Two matrices generate monomially equivalent codes if one can be obtained from the other by left multiplication by an invertible 4×4 matrix and right multiplication by a monomial matrix (permutation and scaling of columns). The number of equivalence classes is something. But given one code C, how many matrices generate a code in the same equivalence class? That is: size of the orbit of C under the action of GL(4,3) × Monomial(7) on the set of 4×7 matrices of rank 4. The stabilizer of C is the automorphism group (monomial automorphisms) which has order 24. The group acting is GL(4,3) × (S_7 ⋉ (F_3^*)^7). But careful: The action: (A, M) * G = A G M, where A in GL(4,3), M is a monomial matrix (7×7). Two generator matrices G1, G2 generate the same code if they are related by left multiplication by an invertible matrix (row operations). Two generator matrices generate equivalent codes if there exists A in GL(4,3) and monomial M such that G2 = A G1 M. The set of all generator matrices that generate a code equivalent to C is the orbit of G under the action of GL(4,3) × Monomial(7). The size of that orbit is |GL(4,3) × Monomial(7)| / |Stab(G)|. The stabilizer consists of pairs (A, M) such that A G M = G (up to maybe scaling? Actually equality as matrices). That's essentially the monomial automorphism group of the code, but also includes left multiplication by matrices that preserve G up to monomial action. Actually, if (A, M) stabilizes G, then A G M = G. This means that A = G M^{-1} G^{-1} (but G is not square). However, the set of such pairs (A, M) is isomorphic to the monomial automorphism group of the code, where an automorphism is a monomial map on column space such that the code is invariant, and then A is determined uniquely by M because rows are a basis. Indeed, for a fixed generator matrix G, if M is a monomial matrix such that G M is row-equivalent to G, i.e., there exists A with A G M = G, then (A, M) is an automorphism. The group of such M (with appropriate A) is the monomial automorphism group. The order given is 24. So |Stab(G)| = 24.

Now, the group GL(4,3) has size (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = 807872*54. Let's compute that.

The monomial group Mon(7) = S_7 ⋉ (F_3^)^7 has size 7! * 2^7 because F_3^ has size 2 (nonzero elements 1 and 2). So |Mon(7)| = 5040 * 128 = 645120.

Thus |GL(4,3) × Mon(7)| = |GL(4,3)| * |Mon(7)|.

The orbit size is |GL(4,3)| * |Mon(7)| / 24.

But the question: "How many generator matrices produce codes that are monomially equivalent to C?" Possibly means: How many 4×7 matrices over F_3 (of rank 4) are there such that the code they generate is monomially equivalent to C? That is exactly the size of the orbit of G under the action described. Because each such matrix generates a code equivalent to C, and conversely any matrix generating a code equivalent to C is in the orbit of some generator matrix of C (since equivalence means there exists A, M such that G' = A G M). But careful: Two different matrices in the orbit might generate the same code? Actually if G' = A G M, then row space of G' is row space of A G M = row space of G M (since A invertible). That is not necessarily equal to row space of G unless M is such that row space of G M equals row space of G, which is exactly the condition for monomial automorphism. So if M is not an automorphism, then row space of G M is a different set of vectors (but equivalent code). So indeed each element of the orbit gives a generator matrix of a code equivalent to C. However, different elements of the orbit could generate the same code? Suppose G1 and G2 are in the orbit, G2 = A G1 M. If they generate the same code, then row space of G1 equals row space of G2. That implies that A G1 M and G1 have same row space, so G1 M and G1 have same row space (since A invertible). That means M is an automorphism of the code generated by G1. But if G1 is in the orbit of G, its automorphism group is conjugate to that of G, same order 24. So the stabilizer of the code (i.e., left multiplication by GL(4,3) alone) is larger. Actually, the action we considered is on matrices. The equivalence classes of codes are orbits under Mon(7) only (since row operations don't change the code). So the number of matrices that generate a given code C is |GL(4,3)| (each ordered basis). The number of matrices that generate any code equivalent to C is |GL(4,3)| times the number of codes equivalent to C. The number of codes equivalent to C is the size of the orbit of C under Mon(7) acting on subspaces. That is |Mon(7)| / |Aut_mon(C)| = 645120 / 24 = 26880. So total matrices = |GL(4,3)| * 26880.

But the question phrasing: "How many generator matrices produce codes that are monomially equivalent to C?" Could be interpreted as: Count all 4×7 matrices over F_3 of rank 4 such that the code they generate is monomially equivalent to C. That's exactly the number we just computed.

Alternatively, maybe they mean: Given the specific generator matrix G, how many generator matrices (i.e., different matrices) generate the same code C? That's just |GL(4,3)|. But then why give automorphism group order? Possibly to compute something else.

Let's read carefully: "How many generator matrices produce codes that are monomially equivalent to C?" The phrase "produce codes that are monomially equivalent to C" suggests we consider matrices that generate a code equivalent to C, not necessarily equal to C. So it's the orbit size under GL(4,3) × Mon(7). But as argued, that orbit size is |GL(4,3)| * |Mon(7)| / 24.

But wait: Does every matrix in the orbit have rank 4? Yes, because A invertible and M invertible. So the orbit is contained in the set of rank-4 matrices. Conversely, any rank-4 matrix G' whose row space is a code equivalent to C: there exists monomial M such that row space of G' M^{-1} equals row space of G (since codes equivalent). Then G' M^{-1} has same row space as G, so there exists A such that G' M^{-1} = A G, so G' = A G M. So indeed G' is in the orbit. So the orbit is exactly the set of all generator matrices of codes equivalent to C.

Thus the number we want is |GL(4,3)| * |Mon(7)| / 24.

Let's compute that using Python.

First compute |GL(4,3)|: (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = (81-1)(81-3)(81-9)(81-27) = 807872*54.

Compute |Mon(7)|: 7! * 2^7 = 5040 * 128.

Then divide by 24.

We'll compute exactly.

[/THOUGHT] [PYTHON] import math

size of GL(4,3)

q = 3 k = 4 gl_size = 1 for i in range(k): gl_size *= (qk - qi) print("|GL(4,3)| =", gl_size)

size of monomial group Mon(7)

mon_size = math.factorial(7) * (2**7) print("|Mon(7)| =", mon_size)

orbit size

orbit_size = gl_size * mon_size // 24 print("Number of generator matrices producing codes monomially equivalent to C =", orbit_size) [/PYTHON]

[THOUGHT] The computed orbit size is 652138905600. This is the number of generator matrices (4×7 over F_3 of rank 4) that produce a code monomially equivalent to C. The answer should be this integer.

[/THOUGHT] \boxed{652138905600}