Berlekamp-Welch Algorithm with Fewer Errors In class we derived how the Berlekamp-Welch algorithm can be used to correct k general errors, given n + 2k points transmitted. In real life, it is usually difficult to determine the number of errors that will occur. What if we have less than k errors? This is a follow up to the exercise posed in the notes. Suppose Alice wants to send 1 message to Bob and wants to guard against 1 general error. She decides to encode the message with Px) 4 (on GF(7)) such that P(0) 4 is the message she want to send. She then sends P(0),P(1),P(2) = (4,4,4) to Bob.

(a) Suppose Bob receives the message (4,5,4). Without performing Gaussian elimination explic- itly, find E(x) and Q(x).
(b) Now, suppose there were no general errors and Bob receives the original message (4,4,4) Show that the Q(x),E(x) that you found in part (a) still satisfies Q(i) E(i) for all i 0,1,2.
(c) Verify that E(x) =x, Q(x) = 4x is another possible set of polynomials that satisfies Q(i) = rE(i) for al 0, 1,2.
q(d) Suppose you're actually trying to decode the received message (4,4,4). Based on what you showed in the previous two parts, what will happen during row reduction when you try to solve for the unknowns?
(e) Prove that no matter what the solution of Cx) and E(x) are though, the recovered P(x) will always be the same.