Sphere packing condition for a perfect code
Proof
Given a codeword
, the number of strings with Hamming distance is given by since there are
combinations of positions different from , and each differing position may be one of letters. Hence, for a code to be perfect, the closed balls of radius must partition , giving the expression above.
The latter part can be interpreted as
The first
terms of a row in Pascal’s triangle sum to a power of .