Problem Analysis #1

[THOUGHT] To count codes equivalent to $C$ under coordinate permutations, compute the orbit size of $C$ under $S_{16}$. By orbit–stabilizer, this is $16!/|\mathrm{Aut}(C)|$. Here $\mathrm{Aut}(C)$ consists of permutations that send each of the 8 coordinate-pairs to a (possibly different) coordinate-pair, with an optional swap inside each pair, so $|\mathrm{Aut}(C)|=2^8\cdot 8!$. I will compute $16!/(2^8\cdot 8!)$ exactly with Python. [/THOUGHT] [PYTHON] import math from fractions import Fraction

val = math.factorial(16) // (28 * math.factorial(8)) val2 = Fraction(math.factorial(16), 28 * math.factorial(8)) print(val) print(val2) print(val == val2) [/PYTHON][THOUGHT] To count codes equivalent to $C$ under coordinate permutations, compute the orbit size of $C$ under $S_{16}$. By orbit–stabilizer, this is $16!/|\mathrm{Aut}(C)|$. Here $\mathrm{Aut}(C)$ consists of permutations that send each of the 8 coordinate-pairs to a (possibly different) coordinate-pair, with an optional swap inside each pair, so $|\mathrm{Aut}(C)|=2^8\cdot 8!$. I will compute $16!/(2^8\cdot 8!)$ exactly with Python. [/THOUGHT] [PYTHON] import math from fractions import Fraction

val = math.factorial(16) // (28 * math.factorial(8)) val2 = Fraction(math.factorial(16), 28 * math.factorial(8)) print(val) print(val2) print(val == val2) [/PYTHON]

[THOUGHT] Using orbit–stabilizer, the number of distinct codes equivalent to $C$ under coordinate permutations is $16!/(2^8\cdot 8!)$, which Python evaluated as 2027025.