lect14 - 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 14: List Decoding Capacity October 2, 2007 Lecturer: Atri Rudra Scribe: Thanh-Nhan Nguyen In the last lecture, we stated a theorem for list decoding capacity, which we restate here: Theorem 0.1 (List-Decoding 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 , ·· ·...
View Full Document

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.

Page1 / 3

lect14 - 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