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Advances in List Decoding of Polynomial Codes

Mrinal Kumar, Noga Ron-Zewi

TL;DR

In this book, significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes are surveyed, including efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms.

Abstract

Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.

Advances in List Decoding of Polynomial Codes

TL;DR

In this book, significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes are surveyed, including efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms.

Abstract

Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.
Paper Structure (109 sections, 45 theorems, 142 equations, 9 figures)

This paper contains 109 sections, 45 theorems, 142 equations, 9 figures.

Table of Contents

  1. Introduction
  2. List decoding
  3. Polynomials in coding theory
  4. List decoding with polynomial codes
  5. Organization and scope of this survey
  6. Notation and definitions
  7. Notations.
  8. List-decodable codes
  9. The Johnson Bound.
  10. The generalized Singleton Bound.
  11. Polynomials over finite fields
  12. Univariate polynomials
  13. Multivariate polynomials
  14. Some families of polynomial codes
  15. Reed-Solomon (RS) Codes.
  16. ...and 94 more sections

Key Result

Theorem 2.1

Let $C \subseteq \Sigma^{n}$ be a code of relative distance $\delta$. Then for any $\alpha<1-\sqrt{1-\delta}$, $C$ is $(\alpha,L)$-list decodable with list size $L = \frac{\delta - \alpha}{(1-\alpha)^{2}-(1-\delta)}$.

Figures (9)

  • Figure 1: Unique decoding of Reed-Solomon Codes
  • Figure 2: List decoding of Reed-Solomon Codes
  • Figure 3: List decoding of Reed-Solomon Codes up to Johnson Bound
  • Figure 4: List decoding of multiplicity codes
  • Figure 5: List decoding of multiplicity codes up to capacity
  • ...and 4 more figures

Theorems & Definitions (122)

  • Theorem 2.1: Johnson Bound
  • proof
  • Theorem 2.2: Generalized Singleton Bound
  • proof
  • Lemma 2.4
  • Lemma 2.5: Schwartz80Zippel79DL78
  • Theorem 2.6: DKSS13, Lemma 2.7
  • Lemma 2.7
  • Lemma 3.1
  • proof
  • ...and 112 more