A code for which the Hamming bound (see Problem 1.17) holds with equality is called a perfect code. a) Show that the repetition code, that is, the rate R = 1/N binary linear code with generator matrix...

A code for which the Hamming bound (see Problem 1.17) holds with equality is called a perfect code.

a) Show that the repetition code, that is, the rate R = 1/N binary linear code with generator matrix G = (1 1 . . . 1), is a perfect code if and only if N is odd.

b) Show that the Hamming codes of are perfect codes. c) Show that the Hamming bound admits the possibility that an N = 23 perfect binary code with ddim = 7 might exist. What must K be?

Problem 1.17

The Hamming sphere of radius t with center at the N-tuple x is the set of allThus, this Hamming sphere contains exactly

distinct N-tuples. Prove the Hamming bound for binary codes, that is,

which is an implicit upper bound on dmin in terms of the block length N and rate R.