1. (a) Let be an (n, k, d) code. Consider a length 2n codebook generated as C' (i,i) : E C). Show that C' is a (2n, k, 2d) code. (b) Let be an (n, k, d) binary code. Consider a length 2n codebook generated as C, = { 1+2) : ' C), where lis a 1 n vector. Show that every codeword in C, has Hamming weight n. (c) The Simplex code is the dual of the Hamming Code, so it has length n = 2k-1, where k is the dimension. Note that the Simplex code has a generator matrix whose columns consists of all non-zero k-length binary vectors. Use the results of part (a), (b) to show that the Simplex code has minimum distance 2k-1 Hint: Use induction. 1. (a) Let be an (n, k, d) code. Consider a length 2n codebook generated as C' (i,i) : E C). Show that C' is a (2n, k, 2d) code. (b) Let be an (n, k, d) binary code. Consider a length 2n codebook generated as C, = { 1+2) : ' C), where lis a 1 n vector. Show that every codeword in C, has Hamming weight n. (c) The Simplex code is the dual of the Hamming Code, so it has length n = 2k-1, where k is the dimension. Note that the Simplex code has a generator matrix whose columns consists of all non-zero k-length binary vectors. Use the results of part (a), (b) to show that the Simplex code has minimum distance 2k-1 Hint: Use induction