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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 01/24/2012.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online