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**Unformatted text preview: **SySc 645 Information Theory: Final Exam. Solutions 1. (30 points) A Hamming code with block length 7 can be characterized by the matrix H = 1 1 1 1 1 1 1 1 1 1 1 1 Each legal input codeword x satisfies Hx = 0 (all arithmetic is understood to be mod 2). The 7 bits received can be represented by y = x + e where e is the error pattern. Assume that the bits are transmitted over a memoryless binary symmetric channel with the probability of error in each bit of which is much less than 1 2 . (a) How many codewords (legal input x s) are there? Answer: 16 (b) If y = 1110010 what is x ( y ), the best guess for x ? Answer: 1110000 (c) Assuming that you attempt to correct errors in received words, find P E , the probability that you will make an error in guessing the trans- mitted codeword: i. First write out an exact algebraic expression. Answer: P E = 1- (1- ) 7- 7 (1- ) 6 ii. Give an approximate numerical value for the expression assuming that = 10- 9 ....

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