Let be the binary linear code
How many binary linear codes are there which are equivalent (under coordinate permutations) to ?
[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}