# lect13 - Error Correcting Codes Combinatorics Algorithms...

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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: Thanh-Nhan Nguyen & Atri Rudra In previous lectures, we have seen the following bound for unique decoding (for worst-case 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 worst-case 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 ≤ ρ ≤ 1 ,L ≥ 1 , a code C ⊆ Σ n is ( ρ,L )-list decodable if for every received word y ∈ Σ n , |{ c ∈ C | Δ( y ,c ) ≤ ρn }| ≤ L Given an error parameter ρ , a code C and a received word y , a list-decoding algorithm should output all codewords in C that are within (relative) Hamming distance ρ from y . Note that if the fraction of errors that occurred during transmission is at most ρ then the transmitted codeword is guaranteed to be in the output list. Further, note that if C is ( ρ,L )-list decodable then the algorithm will always output at most...
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lect13 - Error Correcting Codes Combinatorics Algorithms...

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