Problem Analysis #1

[THOUGHT] To determine the number of binary linear codes equivalent to $C$ under coordinate permutations, I need to count the distinct ways to permute the coordinates of $C$ while preserving its structure. The code $C$ consists of 8 pairs of identical bits, so any equivalent code must also have this structure of 8 pairs.

The problem can be seen as equivalent to the number of ways to partition the 16 coordinates into 8 unordered pairs. This is a classic combinatorial problem. The formula for the number of ways to partition $2n$ items into $n$ unordered pairs is given by:

For $n = 8$, we get:

I will calculate this using Python to ensure accuracy. [/THOUGHT] [PYTHON] import math

n = 8 result = math.factorial(2*n) // (2**n * math.factorial(n)) print(result)

[THOUGHT] Based on the Python output, the number of binary linear codes equivalent to $C$ under coordinate permutations is 2027025. This confirms the calculation of the number of ways to partition 16 coordinates into 8 unordered pairs. [/THOUGHT] \boxed{2027025}