Problem Analysis #7

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5,418 toks

Problem Statement

Let

G = \begin{pmatrix} 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 \end{pmatrix}

be a generator matrix of a ternary linear code $C \subset \mathbb{F}_3^7$ . It is known that the order of the group of monomial automorphisms of $C$ is 24. How many generator matrices produce codes that are monomially equivalent to $\mathcal{C}$ ?\

Ground Truth

652138905600

Execution Trace

Reasoning Process

import math

def gl_order(q, n):
    return math.prod((q**n - q**i) for i in range(n))

# Calculate order of GL(7,3)
gl = gl_order(3, 7)
aut_group_size = 24
num_equivalent_matrices = gl // aut_group_size
num_equivalent_matrices

Tool Output

[PYTHON OUTPUT] 5586185175111586805760 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The number of generator matrices producing codes monomially equivalent to C is equal to the order of the general linear group GL(7, 3) divided by the order of the automorphism group (24). Using the previously determined GL(7, 3) value of 5586185175111586805760, divide it by 24 to get the final answer. [/THOUGHT] [PYTHON] num_equivalent_matrices = 5586185175111586805760 // 24 num_equivalent_matrices [/PYTHON][THOUGHT] The number of generator matrices producing codes monomially equivalent to C is equal to the order of the general linear group GL(7, 3) divided by the order of the automorphism group (24). Using the previously determined GL(7, 3) value of 5586185175111586805760, divide it by 24 to get the final answer. [/THOUGHT] [PYTHON] num_equivalent_matrices = 5586185175111586805760 // 24 num_equivalent_matrices [/PYTHON]

Tool Output

[PYTHON OUTPUT] 232757715629649450240 [/PYTHON OUTPUT] [PYTHON OUTPUT] 232757715629649450240 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] We have confirmed through two consecutive computations that dividing the order of GL(7,3) by 24 yields the number of equivalent generator matrices. Both runs produced identical results: 232757715629649450240. This confirms the accuracy of our calculation. [/THOUGHT]

Final Answer: \boxed{232757715629649450240}[THOUGHT] We have confirmed through two consecutive computations that dividing the order of GL(7,3) by 24 yields the number of equivalent generator matrices. Both runs produced identical results: 232757715629649450240. This confirms the accuracy of our calculation. [/THOUGHT]

Final Answer: \boxed{232757715629649450240}

Final Model Output

232757715629649450240

Comparative Analysis (P7) (Average across runs)

Avg Time Taken

Qwen3-30B-A3B-2507

1905.6

Gemma-3-27B

873.1

Deepseek-R1-distill-Qwen-32B (high)

168.1

Gemini-3-Pro-Preview

76.1

EXAONE-4.0.1-32B (high)

74.4

Grok-4.1-fast

71.4

GPT-oss-20B (high)

56.1

Solar-Pro-2 (31B)(high)

55.1

Deepseek-V3.2

41.8

HCX-007(high)

35.0

GPT-5.1 (high)

25.1

Llama-VARCO-8B-Instruct

24.7

Claude-Opus-4.5

17.4

A.X-4.0 (72B)

13.5

Avg Token Usage

Gemini-3-Pro-Preview

13535.7

Grok-4.1-fast

11559.7

Qwen3-30B-A3B-2507

10081.0

Solar-Pro-2 (31B)(high)

9138.0

GPT-oss-20B (high)

8720.3

Deepseek-V3.2

5870.7

HCX-007(high)

5705.3

Deepseek-R1-distill-Qwen-32B (high)

4983.0

EXAONE-4.0.1-32B (high)

4507.3

Claude-Opus-4.5

4192.7

GPT-5.1 (high)

4129.3

Llama-VARCO-8B-Instruct

3571.3

A.X-4.0 (72B)

3021.3

Gemma-3-27B

2368.7