Problem Analysis #1

[THOUGHT] The number of binary linear codes equivalent to C under coordinate permutations is the size of the orbit of C under the action of the symmetric group S_16. This is given by |S_16| / |Stab(C)|. The stabilizer Stab(C) consists of permutations preserving the partition of coordinates into 8 pairs (each pair identical in C), which has order 2^8 * 8!. Thus, the count is 16! / (2^8 * 8!). [/THOUGHT] [PYTHON] import math n = 16 k = 8 result = math.factorial(n) // (2**k * math.factorial(k)) print(result) [/PYTHON]

[OBSERVATION] 2027025 [/OBSERVATION]

[THOUGHT] The result of the computation is 2027025. [/THOUGHT] $\boxed{2027025}$

[THOUGHT] The Python output confirms the result is 2027025. [/THOUGHT] $\boxed{2027025}$