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# Exam2 - University of Illinois Spring 2011 ECE 361 Second...

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University of Illinois Spring 2011 ECE 361: Second MidSemester Exam Tuesday April 12, 2010, 9:30 a.m. – 10:50 a.m. 1. Let x , y , and z denote 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.

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• Spring '09
• Elementary arithmetic, Even and odd functions, Parity, Evenness of zero, conditional error probability, standard discrete-time model

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Exam2 - University of Illinois Spring 2011 ECE 361 Second...

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