Let
be a generator matrix of a ternary linear code . It is known that the order of the group of monomial automorphisms of is 24. How many generator matrices produce codes that are monomially equivalent to ?\
[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.
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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 ).
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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.
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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 ).
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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)| ).
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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 ).
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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.