The question is about coding theory. Please prove the following question as many as possible. Thanks!!

Let C be a binary code of length n. Let A = {0, 1}. (a) Let 7 be a permutation of {1, ..., n}, i.e. 7 : {1, ...,n} -> {1, ..., n} is a bijection. Let T : A" - A" defined by x1 . . . an > CT(1) . . .CT(n). Let "' = T(C), show that d(C') = d(C). (b) Let c = C1 . .. Cn E An. Let U : An - An defined by x1 ... an - (a1+ C1, . . ., Un + Cn). Let C' = U(C). Show that d(C') = d(C). (c) Let C be a binary (n, M, d)-code. Show that there is a (n, M, d)-code containing 0. . .0 and 1 . .. 10 ...0. d n-d