hw2_810

# hw2_810 - 0 and&amp;lt 1-p there is a binary code C of...

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Math 810, Spring 2011 – HW 2 Due Monday, 21 February 2011 As before, you may talk with each other about problems but make sure you write up your solutions separately. Also always explain things carefully and thoroughly. 1. Problem 3.1.3 from the Notes : http://www.math.msu.edu/~jhall/classes/codenotes/coding-notes.html 2. Problem 3.1.8 from the Notes : 3. Problem 3.1.9 from the Notes . 4. We now have Shannon’s Theorem on the BEC ( p ) with p [0 , 1) : ( * ) for every > 0 and κ < 1 - p there is a binary code C of rate at least κ and a decoding algorithm A ( C ) such that avg x C P x A ( C ) ( C ) < . Use ( * ) to prove the stronger ( ** ) for every >
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Unformatted text preview: 0 and &amp;lt; 1-p there is a binary code C of rate at least and a decoding algorithm A ( C ) such that max x C P x A ( C ) ( C ) &amp;lt; . Hint: Choose an appropriate with &amp;lt; &amp;lt; 1-p . Then ( * ) guarantees codes D of rate at least with avg x D P x A ( D ) ( D ) &amp;lt; / 2. A candidate for C would then be the code consisting of those | D | / 2 codewords from D with small values of P x A ( D ) ( D ). (That is, C D is chosen so that for all x C and all z D \ C we have P x A ( D ) ( D ) P z A ( D ) ( D ).)...
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