lect14 - Error Correcting Codes Combinatorics Algorithms...

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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 0 < 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 0 , · · · , m L [ q ] k , tuple ( y , m 0 , · · · , m L ) defines a “bad event” if C ( m i ) B ( y , ρn ) , 0 i L where recall that B ( x , e ) = { z | Δ( x , z ) e } Fix y [ q ] n , m 0 , · · · , m L [ q ] k .
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