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A Refinement of Expurgation

Giuseppe Cocco, Albert Guillén i Fàbregas, Josep Font-Segura

TL;DR

The paper addresses improving error exponents via expurgation for random, pairwise-independent code ensembles. It extends Gallager's expurgation technique by proving that with high probability there exists a mother code of size $(1+\epsilon)M_n$ in which at least $M_n$ codewords attain the expurgated exponent $E_{ex}^{n}(R,Q^n)$ minus a vanishing term, $\delta_n$, as $n$ grows. The result relies on a probabilistic analysis using Markov's inequality, concentration arguments around the expurgated exponent, and a careful choice of subexponential growth sequences $\gamma_n$ to ensure $\delta_n\to 0$. This establishes that good codes — after expurgation of a vanishing fraction — are prevalent in the ensemble, and it strengthens prior existence results by showing high-probability existence of such codes for a broad class of channels, including memoryful ones.

Abstract

We show that for a wide range of channels and code ensembles with pairwise-independent codewords, with probability tending to 1 with the code length, expurgating an arbitrarily small fraction of codewords from a randomly selected code results in a code attaining the expurgated exponent.

A Refinement of Expurgation

TL;DR

The paper addresses improving error exponents via expurgation for random, pairwise-independent code ensembles. It extends Gallager's expurgation technique by proving that with high probability there exists a mother code of size in which at least codewords attain the expurgated exponent minus a vanishing term, , as grows. The result relies on a probabilistic analysis using Markov's inequality, concentration arguments around the expurgated exponent, and a careful choice of subexponential growth sequences to ensure . This establishes that good codes — after expurgation of a vanishing fraction — are prevalent in the ensemble, and it strengthens prior existence results by showing high-probability existence of such codes for a broad class of channels, including memoryful ones.

Abstract

We show that for a wide range of channels and code ensembles with pairwise-independent codewords, with probability tending to 1 with the code length, expurgating an arbitrarily small fraction of codewords from a randomly selected code results in a code attaining the expurgated exponent.
Paper Structure (3 sections, 2 theorems, 17 equations)

This paper contains 3 sections, 2 theorems, 17 equations.

Table of Contents

  1. Preliminaries
  2. Main Result
  3. Proof of Theorem 1

Key Result

Theorem 1

Consider a pairwise-independent code ensemble with $M_n'=M_n(1+\epsilon)$ codewords and any $\epsilon>0$. If the sequence $\{\delta_n\}_{n=1}^\infty$, which depends on the channel and the ensemble, satisfies $\lim_{n\rightarrow\infty}\delta_n=0$, then for any $0<\epsilon_1<\epsilon$, it holds that

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1