Consider the binary symmetric channel: for each bit the transmitter sends, the receiver will receive the same bit with probability p, and will receive the opposite bit with probability (1- p). Without loss of generality, we assume p>0.5. We also assume that the transmitter intends to transmit bit 0 or 1 with probability 0.5 respectively. Let us consider a repetition communication scheme, for which we derive the probability of decoding error for repeating N=1 time or N=3 times. Let us generalize this result to a general positive integer N. (hint: dividing into two cases where N is an even number or odd number)

(1) Please deign a decoding rule for general N, which minimizes the decoding error probability, namely the probability that the decoded bit is not equal to the transmitted bit. Show why this decoding rule minimizes the decoding error probability.

(2) Under such a decoding rule, derive the corresponding minimum decoding error probability.

(3) Show that for N as odd numbers, the minimum decoding error probability strictly decreases, as N increases.

(4) How does the conclusion in (3) change for even-numbered N?