[THOUGHT] To determine the number of binary linear codes equivalent to $C$ under coordinate permutations, we need to count the number of distinct ways to partition the 16 coordinates into 8 unordered pairs. This is equivalent to calculating the number of perfect matchings in a complete graph with 16 vertices. The formula for the number of ways to partition $2n$ elements into $n$ unordered pairs is given by $\frac{(2n)!}{n! \cdot 2^n}$. For $n = 8$, this becomes $\frac{16!}{8! \cdot 2^8}$.

The calculation involves computing the factorials and dividing appropriately. Using Python, we can verify this result by directly computing the value. </think>

[PYTHON] import math

Calculate the number of equivalent codes

numerator = math.factorial(16) denominator = math.factorial(8) * (2 ** 8) result = numerator // denominator print(result) [/PYTHON]

[THOUGHT] The Python output confirms the calculation. The number of binary linear codes equivalent to $C$ under coordinate permutations is $\frac{16!}{8! \cdot 2^8} = 2027025$. [/THOUGHT] $\boxed{2027025}$