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"""Gray code — generate n-bit Gray code sequences.

A Gray code is an ordering of binary numbers such that successive values

differ in exactly one bit. Used in error correction and rotary encoders.

Inspired by PR #932 (Simranstha045).

"""

from __future__ import annotations

def gray_code(n: int) -> list[int]:

"""Return the n-bit Gray code sequence as a list of integers.

Uses the reflection (mirror) construction:

gray(i) = i ^ (i >> 1)

>>> gray_code(2)

[0, 1, 3, 2]

>>> gray_code(3)

[0, 1, 3, 2, 6, 7, 5, 4]

"""

return [i ^ (i >> 1) for i in range(1 << n)]

def gray_to_binary(gray: int) -> int:

"""Convert a Gray-coded integer back to standard binary."""

mask = gray >> 1

while mask:

gray ^= mask

mask >>= 1

return gray