Problem Analysis #1

[THOUGHT] The code $C$ has a specific structure where each $a_i$ appears twice consecutively. To find the number of binary linear codes equivalent to $C$ under coordinate permutations, we need to consider all possible distinct arrangements of the coordinates that preserve the pairing structure of $C$.

The key observation is that the code $C$ is defined by pairs of identical coordinates. The equivalence under coordinate permutations means we can permute the 16 coordinates, but the resulting code must still have the same structure: pairs of identical coordinates.

The number of distinct codes equivalent to $C$ is equal to the number of ways to partition the 16 coordinates into 8 ordered pairs, where the order within each pair does not matter (since swapping the two identical elements in a pair does not change the code).

This is equivalent to counting the number of ways to choose 8 pairs from 16 coordinates, which is given by the multinomial coefficient $\frac{16!}{(2!)^8 \cdot 8!}$, since we divide by $2!$ for each pair (order within the pair does not matter) and $8!$ (order of the pairs does not matter).

[/THOUGHT] [PYTHON] from math import factorial

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

[THOUGHT] The Python output confirms the calculation: there are 2,027,025 distinct binary linear codes equivalent to $C$ under coordinate permutations. [/THOUGHT]

$\boxed{2027025}$