2. (25 pts) Repetition Code and BSC Consider the binary symmetric channel (BSC) and repetition code described in Lecture 6. The BSC has probability p to flip a bit sent through this channel, with 0 < p < 0.5. To transmit information reliably through this channel, we use the N repetition code which repeats the bit we want to transmit N times (known as encoding), and then transmits this N bit sequence through the BSC. Upon receiving this sequence, the receiver uses the majority rule to decide the encoded bit (this is decoding). For example, for N = 5, suppose we want to transmit a 0. We first encode it into the sequence 00000, using the 5-repetition code. Suppose some bits got flipped during transmission over this BSC, so that the receiver observes the sequence 00011. In this example, using majority vote, the receiver would determine that the encoded bit for transmission was a 0. In this question, assume that the bits encoded for transmission are equiprobable. (a) Write a program to simulate the BSC. Also write programs for encoding (repe- tition code) and decoding (majority rule). We choose N 1, 3,5 and 7 for this problem. For a given N, simulate 20000 instances of sending a bit through the BSC using repetition code and record the fraction of error, i.e., empirical error probability, for p {0.01, 0.02, 0.1}. Plot the empirical error probability against p for each N. You may want to use a semi-log for the y axis. (b) Find the theoretical error probability of sending a bit through BSC using repeti- tion code, parameterized by N and p. For N 1, 3, 5 aud 7 and p {0.01, 0.02, , 0.1}, generate plots similar to (a) using your theoretical results. (c) What kind of channel would be a BSC with p = 0.5 ? What change, if any, would you employ in your system design if you knew that 0.5 < p < 1 ?