Suppose that the codewords 11000 and 01111 are received across a noisy radio channel. The code is C := {00000, 01101, 10110, 11011} . Which of the two received messages are we more confident about decoding without requesting retransmission? Is it 100% certain that we will decode at least one message correctly?

(a) Explain briefly why two equivalent q-ary codes have the same parameters (n. M. d). (b) Are two codes with the same parameters (n. M, d) necessarily equivalent? Either prove it in general, or disprove it with an explicit counterexample. (c) Explain why every q-ary code on alphabet {0, 1,2,...q-1} is equivalent to a code containing the codeword 000...0. (d) Using c), or otherwise, determine the unique binary (5, M, 3)-code with M equal to A (5,3), up to equivalence. Describe the main steps in deducing the answer. What is the value of M? (e) Determine the Gilbert-Varshamov and Hamming bounds on A (5,3). Was the code you found in d) a perfect code?