midterm-solution - University of California San Diego Fall...

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University of California San Diego Fall 2008 ECE 259A: Solutions to the Midterm Exam Problem 1. a. Assume to the contrary that some of the vectors have even weight and some have odd weight. Specifically, suppose we have x , y , z in our set, with wt x , wt y even and wt z odd. Then d x , y wt x wt y 2wt x y 0 mod 2 (1) d x , z wt x wt z 2wt x z 1 mod 2 (2) where x y denotes a vector that is nonzero only in those positions where both x and y are non- zero. This is a contradiction, since d x , y d x , z d . b. It is obvious from (1) that the distance between any two vectors of the same parity (that is, either both even weight or both odd weight) is even. Hence d must be even by (a). c. Without loss of generality, let 0 , x , y be the three vectors. Then wt x wt y d , in con- junction with d x , y wt x wt y 2wt x y d , implies that wt x y d 2 . Let u x y . Since wt u d 2 , it is clear that u is at distance d 2 from 0 , x , y . It remains to show that such a vector u is unique. Assume to the contrary that there exists another vector v at distance d 2 from 0 , x , y . Then
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