lect29 - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 29: Achieving the BSC p capacity November 6, 2007 Lecturer: Atri Rudra Scribe: Yang Wang & Atri Rudra 1 Derandomized GMD algorithm We introduced the GMD algorithm in the last lecture. Recall that we presented two randomized versions of the algorithm last time. Today we will present the derandomized version. Note that last time we proved that there exists a value θ [0 , 1] such that the decoding algorithm works correctly. Obviously we can obtain such a θ by doing an exhaustive search for θ . Unfortunately, there are uncountable choices of θ because θ [0 , 1] . However, this problem can be taken care of by the standard discretization trick. Define Q = { 0 , 1 } ∪ { 2 w 1 d , · · · , 2 w N d } . Then because for each i , w i = min(Δ( y 0 i , y i ) , d/ 2) , we have Q = { 0 , 1 } ∪ { q 1 , · · · , q m } where q 1 < q 2 < · · · < q m for some m ≤ b d 2 c . Notice that for every θ [ q i , q i +1 ) , the Step 1 of the second version of GMD algorithm outputs the same y 00 . Thus, we need to cycle through all possible value of θ Q , leading to the following algorithm: Input : y
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