# ex5_sol - Introduction to Information Theory(67548...

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Introduction to Information Theory (67548) January 27, 2009 Assignment 5- Solution Lecturer: Prof. Michael Werman Due: Note: Unless speciﬁed otherwise, all entropies and logarithms should be taken with base 2 . Problem 1 Another Error Correcting Codes Bound 1. Assume we have three diﬀerent code words, x 1 ,x 2 ,x 3 . Without loss of generality, we may assume that the ﬁrst 2 n/ 3 bits of x 1 ,x 2 are all 0’s and all 1’s respectively. Now, consider codeword x 3 . Regardless of the values of its last n/ 3 bits, it has to have at least n/ 3 + 1 1’s in its ﬁrst 2 n/ 3 entries, in order to have a distance of more than 2 n/ 3 from x 1 . But this implies that the distance of x 3 and x 2 is less than 2 n/ 3, a contradiction. Therefore we can have at most 2 codewords (say the all 0’s and all 1’s bit strings). 2. This is a simple counting argument: the number of ordered pairs x,y C,x 6 = y is | C | ( | C | - 1). For each such pair, d ( x,y ) d , from which the inequality follows. 3. In each column, any pair of a zero and a one contribute 2 to the sum x,y C,x 6 = y d ( x,y ) (1 for the pair (

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ex5_sol - Introduction to Information Theory(67548...

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