[THOUGHT] Let me analyze this problem step by step.

We have a generator matrix G for a ternary linear code C ⊂ F_3^7. The code has dimension k=4 and length n=7.

A generator matrix is in standard form [I_k | P] where I_k is the k×k identity matrix.

Two codes are monomially equivalent if one can be obtained from the other by:

1. Permuting the coordinates (columns)
2. Multiplying each coordinate by a non-zero scalar

The group of monomial automorphisms of C has order 24.

The number of generator matrices that produce codes monomially equivalent to C can be computed as follows:

The total number of ways to choose a generator matrix for any code monomially equivalent to C is:

• Total monomial transformations on F_3^7: The monomial group M_n(F_3) has order n! × (q-1)^n = 7! × 2^7 for n=7 and q=3
• But we need to count generator matrices, not just codes

Actually, let me think more carefully. A generator matrix G produces a code C. Two generator matrices G1 and G2 produce monomially equivalent codes if and only if there exists a monomial matrix M such that the row space of G1·M equals the row space of G2.

The number of generator matrices for a fixed code C of dimension k over F_q with length n is:

• The number of ordered bases of a k-dimensional vector space over F_q
• This is (q^k - 1)(q^k - q)(q^k - q^2)...(q^k - q^(k-1)) = |GL_k(F_q)|

For k=4 and q=3: |GL_4(F_3)| = (3^4-1)(3^4-3)(3^4-9)(3^4-27) = 80 × 78 × 72 × 54

The monomial group M_7(F_3) has order 7! × 2^7 = 5040 × 128 = 645120

The number of codes monomially equivalent to C is |M_7(F_3)| / |Aut(C)| = 645120 / 24 = 26880

The number of generator matrices producing codes monomially equivalent to C is: (Number of equivalent codes) × (Number of generator matrices per code) = 26880 × |GL_4(F_3)|

Let me compute this. [/THOUGHT] [PYTHON] import math

Parameters

n = 7 # length k = 4 # dimension q = 3 # field size (ternary)

Order of GL_k(F_q) - number of generator matrices for a fixed code

gl_k_q = 1 for i in range(k): gl_k_q *= (qk - qi) print(f"|GL_4(F_3)| = {gl_k_q}")

Order of monomial group M_n(F_q)

monomial_group_order = math.factorial(n) * (q-1)**n print(f"|M_7(F_3)| = {monomial_group_order}")

Order of automorphism group of C

aut_C = 24 print(f"|Aut(C)| = {aut_C}")

Number of codes monomially equivalent to C

num_equivalent_codes = monomial_group_order // aut_C print(f"Number of equivalent codes = {num_equivalent_codes}")

Number of generator matrices producing codes monomially equivalent to C

num_generator_matrices = num_equivalent_codes * gl_k_q print(f"Number of generator matrices = {num_generator_matrices}") [/PYTHON]

[THOUGHT] The calculation shows that:

• |GL_4(F_3)| = 24261120 (number of generator matrices for a single code)
• |M_7(F_3)| = 645120 (order of the monomial group)
• Number of codes monomially equivalent to C = 645120/24 = 26880
• Total number of generator matrices = 26880 × 24261120 = 652138905600 [/THOUGHT]

\boxed{652138905600}