Problem 3. (23 points) In this problem, we use the term perfect code to refer specically to an (n, M, d) binary code (either linear or nonlinear) with n = 2m 1, M = 2"_m and d = 3, for an integer m 2 1. An extended perfect code C* is a code of length n + 1 = 2"\" obtained from a perfect code C of length n by appending an extra coordinate, so that the weight of all the codewords in CC\" is even. a. Let C C 1F;1 be a perfect code. Prove that there exists a partition C0, C1, . . . , CH of 13'2" into perfect codes, where C0 = C and C,- = 114+ C is a translate of C for i = 1,2,. . ., n. b. Let 1E;1+1 be the set of all even weight vectors in 1F;1+1 . Show that, given an extended perfect code C*C 1311:\