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Unformatted text preview: CSCI 415 Computer Networks Homework 3 Solution Saad Mneimneh Computer Science Hunter College of CUNY Problem 1: Error correcting codes Suppose that a parity check code has a minimum distance d . Deﬁne the distance of a code word and a received string (of the same length) as the number of bit positions in which the two are diﬀerent (that’s the Hamming distance). (a) Show that if the distance between a code word and a given string is less than d/ 2, the distance between any other code word and the given string must exceed d/ 2. ANSWER : By contradiction: Let x denote the string, and y and z be two codewords. If the distance between x and y is less than d/ 2, and the distance between x and z is less than or equal to d/ 2, then the distance between y and z is less than d , contradicting the assumption that the code has a minimum distance of d . (b) Show that if a decoder maps a given string into a code word at smallest dis-tance from the string, all combinations of fewer than d/ 2 erros will be corrected. ANSWER : Let x be the string, and y be the codeword with minimum distance d to that string. Because fewer than d/ 2 errors occurred, d < d/ 2. From part (a), such codeword y is unique. Therefore, y must be the codeword that was transmitted. transmitted....
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This note was uploaded on 03/27/2010 for the course CSCI 415 taught by Professor Saadmneimneh during the Spring '08 term at CUNY Hunter.
- Spring '08
- Computer Networks