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Continuous Optimization for Decoding Errors

Shashank Srivastava

TL;DR

A list decoding framework that reduces the task of efficient decoding to proving distance in certain restricted proof systems is described, and an alternate viewpoint of the above framework based on abstract regularity lemmas is given.

Abstract

Error-correcting codes are one of the most fundamental objects in pseudorandomness, with applications in communication, complexity theory, and beyond. Codes are useful because of their ability to support decoding, which is the task of recovering a codeword from its noisy copy. List decoding is a relaxation where the decoder is allowed to output a list of codewords, and has seen tremendous progress over the last 25 years. In this thesis, we prove new algorithmic and combinatorial results about list decoding. We describe a list decoding framework that reduces the task of efficient decoding to proving distance in certain restricted proof systems. We then instantiate this framework for Tanner codes of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] and Alon-Edmonds-Luby (AEL) distance amplification [FOCS 1995] of unique decodable base codes to get the first polynomial time list decoding algorithms for these codes. We also show extensions to the quantum version of AEL distance amplification to get polynomial-time list decodable quantum LDPC codes. We next give an alternate viewpoint of the above framework based on abstract regularity lemmas. We show how to efficiently implement the regularity lemma for the case of Ta-Shma's explicit codes near the Gilbert-Varshamov bound [STOC 2017]. This leads to a near-linear time algorithm for unique decoding of Ta-Shma's code. We also give new combinatorial results that improve known list sizes beyond the Johnson bound. Firstly, we adapt the AEL amplification to construct a new family of explicit codes that can be combinatorially list decoded to the optimal error correction radius. This is the first example of such a code that does not use significant algebraic ingredients. Secondly, we present list size improvements for Folded Reed-Solomon codes, improving the state of the art list size for explicit list decoding capacity achieving codes.

Continuous Optimization for Decoding Errors

TL;DR

A list decoding framework that reduces the task of efficient decoding to proving distance in certain restricted proof systems is described, and an alternate viewpoint of the above framework based on abstract regularity lemmas is given.

Abstract

Error-correcting codes are one of the most fundamental objects in pseudorandomness, with applications in communication, complexity theory, and beyond. Codes are useful because of their ability to support decoding, which is the task of recovering a codeword from its noisy copy. List decoding is a relaxation where the decoder is allowed to output a list of codewords, and has seen tremendous progress over the last 25 years. In this thesis, we prove new algorithmic and combinatorial results about list decoding. We describe a list decoding framework that reduces the task of efficient decoding to proving distance in certain restricted proof systems. We then instantiate this framework for Tanner codes of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] and Alon-Edmonds-Luby (AEL) distance amplification [FOCS 1995] of unique decodable base codes to get the first polynomial time list decoding algorithms for these codes. We also show extensions to the quantum version of AEL distance amplification to get polynomial-time list decodable quantum LDPC codes. We next give an alternate viewpoint of the above framework based on abstract regularity lemmas. We show how to efficiently implement the regularity lemma for the case of Ta-Shma's explicit codes near the Gilbert-Varshamov bound [STOC 2017]. This leads to a near-linear time algorithm for unique decoding of Ta-Shma's code. We also give new combinatorial results that improve known list sizes beyond the Johnson bound. Firstly, we adapt the AEL amplification to construct a new family of explicit codes that can be combinatorially list decoded to the optimal error correction radius. This is the first example of such a code that does not use significant algebraic ingredients. Secondly, we present list size improvements for Folded Reed-Solomon codes, improving the state of the art list size for explicit list decoding capacity achieving codes.
Paper Structure (134 sections, 117 theorems, 435 equations, 1 figure, 3 tables, 4 algorithms)

This paper contains 134 sections, 117 theorems, 435 equations, 1 figure, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Basics of Coding Theory
  3. Background and Motivation
  4. Overview of Results
  5. A Generic Framework for List Decoding
  6. List Decodable Quantum LDPC Codes
  7. Near-linear time decoding via Regularity
  8. Decodability beyond Johnson Bound
  9. List Size Improvement for Folded Reed-Solomon Codes
  10. Organization
  11. Preliminaries
  12. Reed-Solomon Codes and their Variants
  13. Interleaving of codes
  14. Expander graphs
  15. Graph based codes
  16. ...and 119 more sections

Key Result

Theorem 1.2.1

Let ${\mathcal{C}}$ be a code with distance $\delta$ and alphabet size $q$. Then there is a threshold ${\mathcal{J}}_q(\delta) := (1-1/q) \cdot \left( 1- \sqrt{1- \frac{q}{q-1}\delta}\right) \in \left(\frac{\delta}{2},\delta\right)$ such that the code is combinatorially list decodable upto radius ${

Figures (1)

  • Figure 1: An example of the 1-wide replacement product with outer graph $G = K_5$ and inner graph $H = C_4$. Vertices are labeled by their $H$ components. Note that the rotation map is locally invertible, with $\varphi(1) = 2$, $\varphi(2) = 1$, $\varphi(3) = 4$, and $\varphi(4) = 3$.

Theorems & Definitions (301)

  • Definition 1.1.1: Code, rate and distance
  • Definition 1.1.2: Linear and LDPC codes
  • Theorem 1.2.1: Johnson Bound Joh62G01
  • Definition 1.3.1
  • Lemma 1.3.2: Covering Lemma
  • Corollary 1.3.3: Johnson bound
  • proof
  • Definition 1.3.4: CSS Codes
  • Lemma 1.3.5: Weak Regularity Lemma
  • Corollary 1.3.6: Weak Johnson bound
  • ...and 291 more