I am reading up on perfect code and there's a statement that puzzles me a bit:
We remark that the extended Golay code is not perfect (and indeed cannot be because d is even!)
This makes me wonder, can we make a generate statement: "if d(C) is even, then C is not perfect"? If yes, then what would be the proof for it? Could someone please give some hints?
Let $x$ and $y$ denote two codewords at distance $d$, $d$ even. In $d/2$ places of the $d$ places where $x$ and $y$ differ, flip the bits in $x$ so that they match the bits in $y$. The result is a vector $z$ that is at distance $d/2$ from both $x$ and $y$. How does that jibe with the notion of a perfect code in which the Hamming spheres centered on each codeword are disjoint and fill the space? Which Hamming sphere does $z$ belong to?