University of California San Diego Fall 2008 ECE 259A: Midterm Exam Instructions: There are four problems, weighted as shown below. The exam is open-book: you may use any auxiliary material that you like. Good luck! Problem 1. (25 points) Suppose that M 2 binary vectors of length n are all at Hamming distance d from each other. a. Prove that either all the vectors have even weight, or they all have odd weight. b. Show that d must be even. c. If M 3 , show that there exists a unique vector at distance d 2 from the three vectors. Problem 2. (35 points) Suppose that we wish to communicate over a binary channel that either transmits the codeword at its input as is, or inverts the ±rst i bits in the codeword for some i 1, 2, . . . , n . Thus the set of all possible error-patterns in the channel is n 1 i0 n i : i 0, 1, . . . n , where denotes concatenation. We wish to construct a binary linear code of length n 2 m 1 that corrects all the possible error-patterns: for any c and any e 2 m 1 the codeword
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