[THOUGHT] I will:

1. Verify the rank of G over GF(3) is 4 (so k=4).
2. Compute |GL(4,3)| using the standard formula.
3. Compute size of full monomial group on 7 coordinates over GF(3): (3−1)^7 * 7!.
4. Divide by |Aut(C)|=24 to get the number of distinct monomially equivalent codes.
5. Multiply that by |GL(4,3)| to get total number of generator matrices producing codes monomially equivalent to C. [/THOUGHT] [PYTHON] import itertools import math

Step 1: Check rank of G over GF(3)

G = [ [1,0,0,0,1,2,2], [0,1,0,0,1,1,1], [0,0,1,0,2,1,2], [0,0,0,1,2,0,1] ]

Gaussian elimination over GF(3) to find rank

def rank_mod_p(mat, p): A = [row[:] for row in mat] m, n = len(A), len(A[0]) r = 0 for c in range(n): # find pivot pivot = None for i in range(r, m): if A[i][c] % p != 0: pivot = i break if pivot is None: continue A[r], A[pivot] = A[pivot], A[r] inv = pow(A[r][c], -1, p) # normalize row A[r] = [(xinv) % p for x in A[r]] # eliminate others for i in range(m): if i != r and A[i][c] % p != 0: factor = A[i][c] % p A[i] = [(A[i][j] - factorA[r][j]) % p for j in range(n)] r += 1 if r == m: break return r

rank_G = rank_mod_p(G, 3) print("rank(G) over GF(3) =", rank_G)

Step 2: |GL(4,3)| = product_{i=0..3} (3^4 - 3^i)

q = 3 k = 4 gl_4_3 = 1 for i in range(k): gl_4_3 *= (qk - qi) print("|GL(4,3)| =", gl_4_3)

Step 3: size of full monomial group on 7 coords over GF(3)

n = 7 monomial_group_size = (q-1)**n * math.factorial(n) print("Monomial group size =", monomial_group_size)

Given |Aut(C)| = 24

aut_C = 24

Step 4: number of distinct monomially equivalent codes

num_codes_equiv = monomial_group_size // aut_C print("Number of distinct monomially equivalent codes =", num_codes_equiv)

Step 5: total generator matrices

total_generators = num_codes_equiv * gl_4_3 print("Total generator matrices =", total_generators) [/PYTHON]

[THOUGHT] Using orbit-stabilizer on the full monomial group of size (3−1)^7·7! with stabilizer (automorphism group) of size 24 gives 26880 distinct codes monomially equivalent to C. Each 4-dimensional ternary code has |GL(4,3)|=24261120 ordered bases, hence that many generator matrices. Multiplying yields 652138905600 generator matrices producing codes monomially equivalent to C. [/THOUGHT] \boxed{652138905600}