lect37 - Error Correcting Codes: Combinatorics, Algorithms...

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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 37: List Decoding of RS November 27, 2007 Lecturer: Atri Rudra Scribe: Thanh-Nhan Nguyen & Atri Rudra 1 List Decoding Recap Recall that list decoding capacity is 1- H q ( p ) . In particular, we know that there exists an ( p, O ( 1 ε ))- list decodable code of rate 1- H q ( p )- ε . Here are some natural follow-up questions. Question 1.1. Are there explicit codes that achieve list decoding capacity of 1- H q ( p ) with efficient list decoding algorithms ? Recall that 1- H q ( p ) = 1- p- ε for q = 2 Ω( 1 ε ) , which leads to the following question: Question 1.2. Are there explicit codes with rate R > that can list-decodable up to 1- R- ε faction of errors with efficient list decoding algorithms? Recall that the alphabet-free version of the Johnson bound states that any code with relative distance δ is ( J q ( δ ) , O ( n 2 ))-list decodable, where J q ( δ ) = 1- √ 1- δ. Further, by the Singleton bound, 1- δ ≥ R- o (1) , which in turn implies that J q ( δ ) ≤ 1- √ R. This leads to the following question: Question 1.3. Is there an efficient list decoding algorithm for code of rate R > that can correct 1- √ R fraction of errors? Note that in the question above, explicitness is not an issue as e.g., a Reed-Solomon code of rate R by the Johnson bound is (1- √ R, O ( n 2 ))...
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This note was uploaded on 01/02/2012 for the course CSE 545 taught by Professor Rudra during the Spring '11 term at SUNY Buffalo.

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lect37 - Error Correcting Codes: Combinatorics, Algorithms...

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