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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 5 Linear Binary Coding; Gaussian Channel Capacity Due: Thursday March 10 at 9:30 a.m. Reading: Lecture Notes: Lectures 11 - 14 1. [Distances and Weights] The Hamming distance d H ( x , y ) between two binary vectors x and y is the number of coordinates in which x and y differ. The Hamming weight w H ( x ) of a binary vector x is the number of nonzero entries in x , that is, w H ( x ) = d H ( x , ) where 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. For binary vectors x and y of length n , define x ∧ y as the vector ( x 1 y 1 ,x 2 y 2 ,...,x n y n ). (a) Show that w H ( x ⊕ y ) = w H ( x ) + w H ( y )- 2 · w H ( x ∧ y ). (b) Show that x ⊕ y has even Hamming weight if and only if both x and y have even Hamming weight or both x and y have odd Hamming weight. 2. [Odds and Evens] Let C denote a ( n,k ) linear binary code. Thus, if x and y are codewords in C , then so is their sum x ⊕ y a codeword in C , that is, x ∈ C , y ∈ C ⇒ x ⊕ y ∈ C . (a) Show that for each fixed i , 1 ≤ i ≤ n , it is true that either all the codewords have a 0 in the i-th coordinate, or half of the codewords have a 0 and half have a 1 in the i-th coordinate. That is, either c i = 0 for all codewords c ∈ C or c i = 0 for 2 k- 1 codewords c ∈ C and c i = 1 for the other 2 k- 1 codewords c ∈ C ....
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