[THOUGHT] The code $C$ consists of all vectors in $\mathbb{F}_2^{16}$ where each pair of coordinates $(1,2), (3,4), \dots, (15,16)$ are equal.
Under a permutation $\sigma$ of the coordinates, the image $\sigma(C)$ is determined by the partition of the 16 positions into orbits of $\sigma$.
For $\sigma(C)$ to be equivalent to $C$, every orbit must have size 2, i.e. $\sigma$ must be a perfect matching (a product of 8 disjoint transpositions).
Thus the number of distinct codes equivalent to $C$ equals the number of ways to partition 16 elements into 8 unordered pairs, which is $(16-1)!! = \frac{16!}{2^8\,8!}$.
We will compute this value with Python. [/THOUGHT] [PYTHON] from sympy import factorial

n = 16

Number of perfect matchings (pairings) on n elements = (n-1)!! = (2n)!/(2^n * n!)

num_matchings = factorial(n) // (2**n * factorial(n//2)) print(num_matchings) [/PYTHON]

[THOUGHT] The number of binary linear codes equivalent to C under coordinate permutations is the number of ways to partition 16 coordinates into 8 unordered pairs, which equals (16)! / (2^8 * 8!) = 7918. This matches the Python computation. [/THOUGHT] $\boxed{7918}$