3. Let C be the code over F5 with generator matrix [1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 2 1 3 1 4 = (a) Let ei F be the row vector with a 1 in the th coordinate and 0 everywhere else. Compute the syndrome of e; for each i = 1, ..., 6 using any choice of parity check matrix. (b) How many vectors of weight 1 does F have? Do any of them belong to the same coset of C? (c) Justify that C can correct one error without mentioning anything about its minimum weight. (d) Suppose you receive the following vector: [2 1 3 0 4 0]. Use syndrome decoding to find the original 4-digit message. 3. Let C be the code over F5 with generator matrix [1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 2 1 3 1 4 = (a) Let ei F be the row vector with a 1 in the th coordinate and 0 everywhere else. Compute the syndrome of e; for each i = 1, ..., 6 using any choice of parity check matrix. (b) How many vectors of weight 1 does F have? Do any of them belong to the same coset of C? (c) Justify that C can correct one error without mentioning anything about its minimum weight. (d) Suppose you receive the following vector: [2 1 3 0 4 0]. Use syndrome decoding to find the original 4-digit message