List Decoding of Error-Correcting Codes

1. Front Matter

   PDF

2. 1 Introduction

   1. 1 Introduction

      • Venkatesan Guruswami

      Pages 1-14

   2. 2 Preliminaries and Monograph Structure

      • Venkatesan Guruswami

      Pages 15-30

3. Part I Combinatorial Bounds

   1. Front Matter

      Pages 31-31

      PDF

   2. 3 Johnson-Type Bounds and Applications to List Decoding

      • Venkatesan Guruswami

      Pages 33-44

   3. 4 Limits to List Decodability

      • Venkatesan Guruswami

      Pages 45-78

   4. 5 List Decodability Vs. Rate

      • Venkatesan Guruswami

      Pages 79-92

4. Part II Code Constructions and Algorithms

   1. Front Matter

      Pages 93-93

      PDF

   2. 6 Reed-Solomon and Algebraic-Geometric Codes

      • Venkatesan Guruswami

      Pages 95-145

   3. 7 A Unified Framework for List Decoding of Algebraic Codes

      • Venkatesan Guruswami

      Pages 147-175

   4. 8 List Decoding of Concatenated Codes

      • Venkatesan Guruswami

      Pages 177-207

   5. 9 New, Expander-Based List Decodable Codes

      • Venkatesan Guruswami

      Pages 209-250

   6. 10 List Decoding from Erasures

      • Venkatesan Guruswami

      Pages 251-277

5. Part III Applications

   1. Front Matter

      Pages 279-279

      PDF

   2. Interlude

      • Venkatesan Guruswami

      Pages 281-281

6. Part III Applications

   1. 11 Linear-Time Codes for Unique Decoding

      • Venkatesan Guruswami

      Pages 283-298

   2. 12 Sample Applications Outside Coding Theory

      • Venkatesan Guruswami

      Pages 299-327

   3. 13 Concluding Remarks

      • Venkatesan Guruswami

      Pages 329-332

   4. A GMD Decoding of Concatenated Codes

      • Venkatesan Guruswami

      Pages 333-335

7. Back Matter

   PDF

How can one exchange information e?ectively when the medium of com- nication introduces errors? This question has been investigated extensively starting with the seminal works of Shannon (1948) and Hamming (1950), and has led to the rich theory of “error-correcting codes”. This theory has traditionally gone hand in hand with the algorithmic theory of “decoding” that tackles the problem of recovering from the errors e?ciently. This thesis presents some spectacular new results in the area of decoding algorithms for error-correctingcodes. Speci?cally,itshowshowthenotionof“list-decoding” can be applied to recover from far more errors, for a wide variety of err- correcting codes, than achievable before. A brief bit of background: error-correcting codes are combinatorial str- tures that show how to represent (or “encode”) information so that it is - silient to a moderate number of errors. Speci?cally, an error-correcting code takes a short binary string, called the message, and shows how to transform it into a longer binary string, called the codeword, so that if a small number of bits of the codewordare ?ipped, the resulting string does not look like any other codeword. The maximum number of errorsthat the code is guaranteed to detect, denoted d, is a central parameter in its design. A basic property of such a code is that if the number of errors that occur is known to be smaller than d/2, the message is determined uniquely. This poses a computational problem,calledthedecodingproblem:computethemessagefromacorrupted codeword, when the number of errors is less than d/2.

• Code
• Error-correcting Code
• Information
• Shannon
• algorithms
• coding theory
• complexity theory
• concatenated codes
• decoding
• decoding algorithms
• encoding
• error-detecting codes
• information theory
• list decoding
• reed-solomon codes
• algorithm analysis and problem complexity

• Department of Comp. Sci. & Eng., University of Washington, Seattle, USA

  Venkatesan Guruswami

Policies and ethics