Make a (6, 8, 3) binary code, by trial and error. (Recall this means the code has length 6, minimum distance 3, and 8 codewords.) Hint: You can assume that 000000 is one of the codewords, so all other codewords have weight at least 3. Include as many words of weight 3 as you can, and then try adding in words of weight 4. Look for symmetry patterns to save you from having to check every single pair of codewords.] a. b. Then use the extended version of the Plotkin bound given in class to prove that a (6, 3) code cannot have more than 8 codewords. c. Extend your code from part a. by adding a parity-check bit to each codeword (so that each new codeword has even weight), which should give you a (7, 4) code with 8 codewords. Verify that this code has maximal size by the Plotkin bound, Theorem 3.3. This is "equivalent" to the (7, 8, 4) code made from a Hadamard matrix of order 8 in class, even if it is not obvious! Make a (6, 8, 3) binary code, by trial and error. (Recall this means the code has length 6, minimum distance 3, and 8 codewords.) Hint: You can assume that 000000 is one of the codewords, so all other codewords have weight at least 3. Include as many words of weight 3 as you can, and then try adding in words of weight 4. Look for symmetry patterns to save you from having to check every single pair of codewords.] a. b. Then use the extended version of the Plotkin bound given in class to prove that a (6, 3) code cannot have more than 8 codewords. c. Extend your code from part a. by adding a parity-check bit to each codeword (so that each new codeword has even weight), which should give you a (7, 4) code with 8 codewords. Verify that this code has maximal size by the Plotkin bound, Theorem 3.3. This is "equivalent" to the (7, 8, 4) code made from a Hadamard matrix of order 8 in class, even if it is not obvious