Problem Analysis #1

[THOUGHT] The code $C$ is the set of all vectors in $\mathbb{F}_2^{16}$ whose first 8 coordinates are repeated in the last 8 coordinates: $C=\{(a,a): a\in\mathbb{F}_2^8\}$. Thus $C$ is a 1-dimensional subspace of $\mathbb{F}_2^8$ (all vectors of the form $(a_1,\dots,a_8)$), embedded in $\mathbb{F}_2^{16}$ by duplicating each coordinate. Any coordinate permutation of $C$ must permute the 16 coordinates. Because the code is determined by the 8 “pairs” of equal coordinates, the number of inequivalent permutations is the number of ways to partition the 16 coordinate positions into 8 unordered pairs (a perfect matching), which is

So the number of binary linear codes equivalent to $C$ under coordinate permutations is exactly this value. [/THOUGHT] [PYTHON] from math import factorial

n = 16 k = 8 count = factorial(n) // ( (2**k) * factorial(k) ) print(count) [/PYTHON]

[THOUGHT] The Python output $2027025$ is taken as ground truth, so the number of binary linear codes equivalent to $C$ under coordinate permutations is $\boxed{2027025}$.