The code $C$ is the subspace of $\mathbb{F}_2^{16}$ consisting of all vectors where the coordinates in the pairs $\{1,2\}, \{3,4\}, \dots, \{15,16\}$ are equal. This code is generated by the basis vectors $b_1, \dots, b_8$, where $b_i$ has 1s at positions $2i-1$ and $2i$, and 0s elsewhere.

The group of coordinate permutations is the symmetric group $S_{16}$. Two codes are equivalent if one can be obtained from the other by applying a permutation $\sigma \in S_{16}$ to the coordinates. The number of distinct codes equivalent to $C$ is the index of the automorphism group of $C$ in $S_{16}$, i.e., $|S_{16}| / |\text{Aut}(C)|$.

The automorphism group $\text{Aut}(C)$ consists of all permutations $\sigma \in S_{16}$ such that $\sigma(C) = C$. Since $C$ is the set of vectors constant on the pairs $P_1, \dots, P_8$ (where $P_i = \{2i-1, 2i\}$), any automorphism must map this set of pairs to itself. Specifically:

1. The pairs $P_1, \dots, P_8$ can be permuted among themselves. There are $8!$ such permutations.
2. Within each pair $P_i$, the two elements can be swapped or left alone. There are $2^8$ such combinations. Thus, $|\text{Aut}(C)| = 2^8 \cdot 8!$.

The number of equivalent codes is: [ \frac{|S_{16}|}{|\text{Aut}(C)|} = \frac{16!}{2^8 \cdot 8!} ] This is equal to the product of odd integers from 1 to 15, also known as the double factorial $15!!$.

[PYTHON] import math

def double_factorial(n): if n <= 0: return 1 return n * double_factorial(n - 2)

Calculate 16! / (2^8 * 8!)

numerator = math.factorial(16) denominator = (2**8) * math.factorial(8) result = numerator // denominator

print(f"{numerator=}") print(f"{denominator=}") print(f"{result=}")

Verify with double factorial

result_df = double_factorial(15) print(f"{result_df=}") [/PYTHON] [THOUGHT] The calculated value is 2027025. The logic holds: the number of equivalent codes corresponds to the number of ways to partition the set of 16 coordinates into 8 pairs. The formula is $\frac{16!}{2^8 8!} = 2027025$. [/THOUGHT]

\boxed{2027025}

The number of binary linear codes equivalent to (C) is the number of ways to partition the set of 16 coordinate positions into 8 unordered pairs. This is given by the formula: [ \frac{16!}{8! \cdot 2^8} = 15 \times 13 \times 11 \times 9 \times 7 \times 5 \times 3 \times 1 = 2027025 ]

\boxed{2027025}