Problem Analysis #7

[THOUGHT] The monomial group has order (3-1)^7 * 7! = 2^7 * 5040. The automorphism group order is 24, so orbit size is 2^7 * 5040 / 24. Each code in the orbit has exactly |GL(4,3)| generator matrices, where |GL(4,3)| = (3^4-1)(3^4-3)(3^4-9)(3^4-27). The total number is orbit size times |GL(4,3)|. Compute this exactly using Python. [/THOUGHT] [PYTHON] import math

q = 3 k = 4 n = 7

gl4 = (qk - 1) * (qk - q) * (qk - q2) * (qk - q3) mon_order = (q-1)**n * math.factorial(n) aut_order = 24 orbit_size = mon_order // aut_order total_gens = orbit_size * gl4

print(total_gens) [/PYTHON]

[THOUGHT] The calculation is verified: |GL(4,3)| = 24261120, |M| = 645120, orbit size = 645120 / 24 = 26880, total = 26880 * 24261120 = 652138905600. This is the number of generator matrices for codes monomially equivalent to C. [/THOUGHT] $\boxed{652138905600}$

Explanation

The reasoning follows from group theory and coding theory principles applied to the monomial group action on linear codes.

• The monomial group $M$ for $\mathbb{F}_3^7$ consists of all monomial matrices, which are permutation matrices with nonzero entries on the diagonal scaled by elements of $\mathbb{F}_3^\times = \{1, 2\}$. Its order is $|M| = (3-1)^7 \cdot 7! = 2^7 \cdot 5040 = 128 \cdot 5040 = 645120$.

• The group of monomial automorphisms of $C$, denoted $\text{Aut}_M(C)$, is the stabilizer of $C$ under the action of $M$, with given order 24.

• By the orbit-stabilizer theorem, the size of the orbit of $C$ under $M$ (the number of codes monomially equivalent to $C$) is $|M| / |\text{Aut}_M(C)| = 645120 / 24 = 26880$.

• Each code in this orbit is a 4-dimensional subspace of $\mathbb{F}_3^7$, so the number of generator matrices for any such code equals the number of ordered bases for the row space, which is $|\text{GL}(4, 3)|$.

• Compute $|\text{GL}(4, 3)| = (3^4 - 1)(3^4 - 3)(3^4 - 9)(3^4 - 27) = (81 - 1)(81 - 3)(81 - 9)(81 - 27) = 80 \cdot 78 \cdot 72 \cdot 54 = 24261120$.

• The total number of generator matrices across all codes in the orbit is the orbit size times $|\text{GL}(4, 3)|$: $26880 \cdot 24261120 = 652138905600$.

This matches the verified Python output, confirming the exact count.