Recall that a perfect code can correct up to t errors where the spheres of radius t around codewords are disjoint and cover the entire space. What does this imply about the number of codewords when n = d?

For odd n, consider the Hamming bound (sphere-packing bound) and show that the number of codewords in a perfect code of length n with minimum distance d = n equals the Hamming bound exactly.

Prove that for odd n, the only possible perfect code with d = n is the trivial perfect code where every possible word is a codeword, by showing that any non-trivial code would violate the Hamming bound.