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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 14: List Decoding Capacity October 2, 2007 Lecturer: Atri Rudra Scribe: ThanhNhan Nguyen In the last lecture, we stated a theorem for list decoding capacity, which we restate here: Theorem 0.1 (ListDecoding Capacity) . Let q ≥ 2 be an integer, and < ρ < 1 1 q be a real. (i) Let L ≥ 1 be an integer, there exists an ( ρ,L )list decodable code with rate R ≤ 1 H q ( ρ ) 1 L (ii) For every ( ρ,L ) code of rate 1 H q ( ρ ) + ε , L needs to be exponential in block length of the code. In this lecture, we will prove this theorem. 1 Proof of Theorem 0.1 Proof. We start with the proof of (i). Pick a code C at random where  C  = q k ,k ≤ (1 H q ( ρ ) 1 L ) n. That is, as in Shannon’s proof, for every message m , pick C ( m ) uniformly at random from [ q ] n . Definition 1.1. Given y ∈ [ q ] n , and m , ·· · , m L ∈ [ q ] k , tuple ( y , m , ·· · , m L ) defines a “bad event” if C ( m i ) ∈ B ( y ,ρn ) , ≤ i ≤ L where recall that B ( x ,e ) = { z  Δ( x , z ) ≤ e } Fix y ∈ [ q ] n , m , ·· ·...
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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.
 Spring '11
 RUDRA
 Algorithms

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