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List-Decoding Capacity Implies Capacity on the q-ary Symmetric Channel

Francisco Pernice, Oscar Sprumont, Mary Wootters

TL;DR

It is shown that there is any linear code C⊆ Fqn that has superconstant minimum distance and achieves list-decoding capacity also achieves capacity on the qSC.

Abstract

It is known that the Shannon capacity of the q-ary symmetric channel (qSC) is the same as the list-decoding capacity of an adversarial channel, raising the question of whether there is a formal (and black-box) connection between the two. We show that there is: Any linear code $C\subseteq \mathbb{F}_q^n$ that has minimum distance $d_{\min}=ω(q^3)$ and achieves list-decoding capacity also achieves capacity on the qSC.

List-Decoding Capacity Implies Capacity on the q-ary Symmetric Channel

TL;DR

It is shown that there is any linear code C⊆ Fqn that has superconstant minimum distance and achieves list-decoding capacity also achieves capacity on the qSC.

Abstract

It is known that the Shannon capacity of the q-ary symmetric channel (qSC) is the same as the list-decoding capacity of an adversarial channel, raising the question of whether there is a formal (and black-box) connection between the two. We show that there is: Any linear code that has minimum distance and achieves list-decoding capacity also achieves capacity on the qSC.

Paper Structure

This paper contains 20 sections, 12 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Our question.
  3. Our Results
  4. Overview of Techniques
  5. Related Work
  6. Capacity-Achieving Codes on the qSC$_p$.
  7. List decoding.
  8. Threshold Phenomena.
  9. Conventions and Preliminaries
  10. Finite fields
  11. Probability Theory
  12. Monotonicity
  13. Coding and Decoding
  14. Communication Channels
  15. An isoperimetric inequality over finite fields
  16. ...and 5 more sections

Key Result

Theorem 1

Let $p \in (0,1)$. Let $\{C_n\subseteq\mathbb{F}_{q}^n\}$ be a family of linear codes that achieves list-decoding capacity on the adversarial channel that introduces a $p$-fraction of corruptions. If $d_{\min}(C_n)=\omega(\frac{q^3}{(1-p)^2})$, then $\{C_n\}$ achieves capacity on the qSC$_p$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Remark 1: Requirement on the field size
  • Theorem 3
  • Remark 2: New results for capacity-achieving codes on the qSC$_p$?
  • Lemma 4: Hoeffding's Inequality
  • Definition 1
  • Definition 2
  • Theorem 7
  • proof
  • ...and 18 more