Exam2Soln - University of Illinois Spring 2011 ECE 361:...

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University of Illinois Spring 2011 ECE 361: Second MidSemester Exam: Solutions 1. Let x , y , and z denote three binary vectors of length n . The Hamming distance d H ( x , y ) between x and y is the number of coordinates in which x and y differ. The Hamming weight w H ( x ) of x is the number of nonzero entries in x , that is, w H ( x ) = d H ( x , 0 ) where 0 is the all-zeroes vector. Equivalently, w H ( x ) = i x i where the sum is the ordinary arithmetic sum of 0’s and 1’s, not an XOR summation. (a) Suppose that each of x , y , and z is at the same Hamming distance d > 0 from the others . Mark the box for each true statement among the following. 2 It is possible to find x , y , and z such that w H ( x ), w H ( y ), and w H ( z ) all are odd numbers. 2 It is possible to find x , y , and z such that w H ( x ), w H ( y ), and w H ( z ) all are even numbers. 2 It is possible to find x , y , and z such that exactly two of w H ( x ), w H ( y ), and w H ( z ) are even numbers and the third one is an odd number. 2 It is possible to find x , y , and z such that exactly two of w H ( x ), w H ( y ), and w H ( z ) are odd numbers and the third one is an even number. 2 It is possible to find x , y , and z such that the common distance d is an odd number. 2 It is possible to find x , y , and z such that the common distance d is an even number. Solution: d H ( x , y ) = w H ( x y ) = w H ( x )+w H ( y ) - 2 · w H ( x y ) is even if both w H ( x ) and w H ( y ) are even numbers, or both are odd numbers. If one of w H ( x ) and w H ( y ) is odd and the other is even, then d H ( x , y ) is odd. Thus, if d is odd, then the Hamming weights of x , y , and z must have different parities from each other, which is impossible. In other words, either all three vectors have odd weight or all three have even weight, and in both cases, d must be even. Alternatively, d = w H ( x y ) = w H ( x ) + w H ( y ) - 2 · w H ( x y
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Exam2Soln - University of Illinois Spring 2011 ECE 361:...

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