3. Let C be a q-ary code of length n. Assume the minimal distance d(C) is an odd number, d(C) = 2r + 1. We showed in class that C can always correct up to r errors. That is, whenever a codeword a from C is sent, and r or fewer errors occur in transmission, the Nearest Neighbour Decoding algorithm will decode the received word b correctly (i.e., will decode b as a). Prove that C cannot always correct r + 1 errors. That is, show that there exists a codeword a in C and a word b with d(a, b) = r + 1 such that when a is transmitted and b is received, the Nearest Neighbour Decoding algorithm (either complete or incomplete) will not decode b correctly.
3. Let C be a q-ary code of length n. Assume the minimal distance d(C) is...
1. (30 points) Consider the systematic binary linear (6,3) code with generator matrix 1 0 01 1 0 G- 0 1 0 0 1 1 a) Determine the parity check matrix H of the code. b) What is the minimum distance of the code? How many errors can this code correct and detect? c) Show the results in b) using decoding table d) Find the most likely codeword, given that the noisy received codeword is 010101. e) Now suppose 001101...