hw11sol - Math 55: Discrete Mathematics, Fall 2008 Homework...

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Math 55: Discrete Mathematics, Fall 2008 Homework 11 Solutions * 9. We are to show that every word belongs to the Hamming ball of radius one around some codeword. Since the code corrects 1 error, these Hamming balls are disjoint, so it suffices to show that number of codewords times the size of each Hamming ball is equal to the total number of words, i.e. , 2 m (1 + ± n 1 ² ) = 2 n , where n = 2 k - 1 and m = 2 k - k - 1. The equation holds since 2 2 k - k - 1 (1 + 2 k - 1) = 2 2 k - k - 1 2 k = 2 2 k - 1 11 (a) C = 1 1 1 1 1 1 1 0 1 2 3 4 5 6 0 1 4 2 2 4 1 (b) Compute ³ 2 3 4 ´ C ³ 2 2 3 5 1 5 3 ´ (mod 7). 12. Express the received vector as r = ³ x | y ´ , where x consists of the first m entries of r and y consists of the remaining n - m entries. For any 1 × m vector x , we have x C = ³ x | xP ´ , hence r is a codeword if and only if y = xP . On the other hand, rS = xP - y , so this is equal to zero if and only if the same condition y = xP holds. * 13. Let
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This note was uploaded on 08/16/2010 for the course MATH 55 taught by Professor Strain during the Fall '08 term at Berkeley.

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hw11sol - Math 55: Discrete Mathematics, Fall 2008 Homework...

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