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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 13: List Decoding October 1, 2007 Lecturer: Atri Rudra Scribe: ThanhNhan Nguyen & Atri Rudra In previous lectures, we have seen the following bound for unique decoding (for worstcase errors): p 1 R 2 and the capacity bound for qSC p (for stochastic errors): p H 1 q (1 R ) 1 R (for large q ) . Note that there is a gap between what we can achieve for worstcase errors and stochastic errors. In this lecture, we extend the notion of unique decoding to give the decoder the flexibility to output a list of candidate transmitted codewords. This will allow us to bridge the gap in the Shannon world and the Hamming world. 1 List Decoding The new notion of decoding that we will discuss is called list decoding as the decoder is allowed to output a list of answers. We now formally define (the combinatorial version of) list decoding: Definition 1.1. Given 1 ,L 1 , a code C n is ( ,L )list decodable if for every received word y n , { c C  ( y ,c ) n } L Given an error parameter , a code C and a received word y , a listdecoding algorithm should output all codewords in C that are within (relative) Hamming distance from y . Note that if the fraction of errors that occurred during transmission is at most then the transmitted codeword is guaranteed to be in the output list. Further, note that if C is ( ,L )list decodable then the algorithm will always output at most...
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 Spring '11
 RUDRA
 Algorithms

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