Problem Analysis #7

<|channel|>analysis to=python code<|message|>807872*54

We need to interpret the problem: We have a ternary linear code C of length 7 over F3 with generator matrix G given. The group of monomial automorphisms has order 24. We need to find how many generator matrices produce codes that are monomially equivalent to C.

We have a code C. The question: "How many generator matrices produce codes that are monomially equivalent to \mathcal{C}?" Means: Count the number of generator matrices that generate a code monomially equivalent to C. In other words, how many distinct generator matrices correspond to codes that are monomially equivalent to C? Or maybe: How many generator matrices produce codes that are monomially equivalent to C? That is, how many distinct generator matrices (over F3) produce codes that are monomially equivalent to C? We need to compute the number of generator matrices for all codes that are monomially equivalent to C. But maybe they want the number of generator matrices for all codes that are monomially equivalent to C. That is, how many distinct generator matrices produce a code that is monomially equivalent to C? The group of monomial automorphisms has order 24. That means the size of the stabilizer of C under monomial actions is 24.

We have a code C. The number of distinct codes that are monomially equivalent to C is the size of the orbit of C under the action of the monomial group. The total number of monomial transformations is |Mon(7,3)|? Actually the monomial group on length 7 over F3: It's the set of all 7x7 monomial matrices over F3. A monomial matrix is a permutation matrix times a diagonal matrix with nonzero entries from F3^. Since we are over F3, the multiplicative group F3^ has size 2 (elements {1,2}). So each diagonal entry can be either 1 or 2. So the number of monomial matrices is (number of permutations) * (size of diagonal group)^7 = 7! * 2^7. So |Mon(7,3)| = 5040 * 128 = 645120.

But the group of monomial automorphisms of C has order 24. That means that the stabilizer size is 24. By orbit-stabilizer theorem, the number of distinct monomially equivalent codes to C is |Mon(7,3)| / 24 = 645120 / 24 = 26880. Wait compute: 645120/24 = 26880. So there are 26880 distinct codes monomially equivalent to C.

But the question: "How many generator matrices produce codes that are monomially equivalent to \mathcal{C}?" That might be different: For each code in the orbit, there are many generator matrices that generate it. But maybe they want the number of distinct generator matrices that produce codes monomially equivalent to C? That is, count all generator matrices of dimension 4 (since G has rank 4) that generate a code monomially equivalent to C. But each such generator matrix corresponds to some code in the orbit, but also there are many generator matrices per code. So we need to count all possible generator matrices that produce codes monomially equivalent to C.

We can think: The set of all generator matrices for a given code is the set of all 4x7 matrices that are full rank and whose row space equals C. The number of such generator matrices is equal to the number of invertible 4x4 matrices over F3 times something? Actually, given a fixed code C of dimension k=4, the number of generator matrices for it is equal to |GL(k,F3)| * something? Wait, we need to count all 4x7 matrices whose row space equals C. For a given code, the set of generator matrices is all kx n matrices whose rows span C. Each such matrix can be obtained by taking a basis of C and writing it as a 4x7 matrix. The number of bases of a k-dimensional vector space over F_q is equal to the number of invertible kxk matrices times something? Actually, the number of ordered bases is |GL(k,F_q)|. But each generator matrix corresponds to an ordered basis (the rows). So the number of generator matrices for a given code is |GL(k,F_q)|. Because each ordered basis yields a generator matrix; conversely, any generator matrix gives an ordered basis of the code. So for each code C (dimension 4), there are |GL(4,3)| generator matrices. Let's compute that: |GL(4,3)| = (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3). Compute: 3^4 = 81. So:

• (81 - 1) = 80
• (81 - 3) = 78
• (81 - 9) = 72
• (81 - 27) = 54

Multiply: 80 * 78 * 72 * 54. Let's compute that.<|channel|>commentary to=python code<|message|>807872*54

<|channel|>commentary to=python code<|message|>26880 * 24261120

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