hw11sol

# 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 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 2 3 4 C 2 2 3 5 1 5 3 (mod 7). 12. Express the received vector as r = 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 = 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 E ( x ) = x + u , Q ( x ) = ax 3 + bx 3 + cx + d . The key equations E ( i ) = R i Q ( i ) (mod 11) for i = 0 , 1 , 2 , 3 , 4 give 10 d + 9 u = 0 10 a + 10 b + 10 c + 10 d + 2 u + 2 = 0 3 a + 7 b + 9 c + 10 d

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