lect13 - Error Correcting Codes: Combinatorics, Algorithms...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 list-decoding 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...
View Full Document

Page1 / 3

lect13 - Error Correcting Codes: Combinatorics, Algorithms...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online