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The Expurgated Error Exponent is Not Universally Achievable

Seyed AmirPouya Moeini, Marco Dalai, Albert Guillén i Fàbregas

TL;DR

A family of DMCs is constructed for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero.

Abstract

We study the universal attainability of the expurgated error exponent for discrete memoryless channels (DMCs). While the random-coding exponent is known to be universally attainable via maximum mutual information (MMI) decoding for DMCs, it remains open whether the expurgated exponent can be attained universally. We show that this is not the case in general. Specifically, we construct a family of DMCs for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero. In addition, for the same channel family, we show that MMI decoding fails to achieve the expurgated exponent for any channel in the family.

The Expurgated Error Exponent is Not Universally Achievable

TL;DR

A family of DMCs is constructed for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero.

Abstract

We study the universal attainability of the expurgated error exponent for discrete memoryless channels (DMCs). While the random-coding exponent is known to be universally attainable via maximum mutual information (MMI) decoding for DMCs, it remains open whether the expurgated exponent can be attained universally. We show that this is not the case in general. Specifically, we construct a family of DMCs for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero. In addition, for the same channel family, we show that MMI decoding fails to achieve the expurgated exponent for any channel in the family.
Paper Structure (6 sections, 6 theorems, 57 equations, 2 figures)

This paper contains 6 sections, 6 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Counterexample
  4. A Counterexample for MMI Decoding
  5. Proof of Theorem \ref{['thm:comp']}
  6. Rate zero proof for binary symmetric channels

Key Result

Theorem 1

There exists a family of DMCs $\mathcal{W}$ and a rate threshold $R_{\mathcal{W}}>0$ with the following property. For any channel $W\in\mathcal{W}$, any $\delta>0$, and any rate $R\in[0,R_{\mathcal{W}})$, consider an arbitrary sequence of block codes ${(f_n,\varphi_n)}$ of rate $R$ satisfying for all sufficiently large $n$. Then, one can find another channel $\widehat{W} \in \mathcal{W}$ for whic

Figures (2)

  • Figure 1: The converse exponent $-\log\!\left(\tfrac{1-\varepsilon}{2}\right)$, the rate-zero expurgated exponent $-\frac{1}{2}\log\!(2\sqrt{\varepsilon(1-\varepsilon)})$, and the rate-zero random-coding exponent $-\log\!(\tfrac{1}{2}+\sqrt{\varepsilon(1-\varepsilon)})$ for the channels $W_{\varepsilon}$ and $\widehat{W}_{\varepsilon}$.
  • Figure 2: Comparison between the expurgated error exponent and the converse exponent $-\log\![(1-\varepsilon)/2]$ for $\varepsilon = 0.001$. In this example, ${R}_{\varepsilon} \approx 0.1865$, so any code of rate below this value cannot attain the expurgated exponent under MMI decoding.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 4
  • proof