Question: 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

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:\

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