List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition / Edition 1

by Venkatesan Guruswami

Overview

How can one exchange information effectively 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 efficiently. This thesis presents some spectacular new results in the area of decoding algorithms for error-correctingcodes. Specifically, it shows how the notion of “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. Specifically, 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, called the decoding problem: compute the message from a corrupted codeword, when the number of errors is less than d/2.

Product Details

ISBN-13:	9783540240518
Publisher:	Springer Berlin Heidelberg
Publication date:	05/01/2005
Series:	Lecture Notes in Computer Science , #3282
Edition description:	2005
Pages:	352
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 Introduction.- 1 Introduction.- 2 Preliminaries and Monograph Structure.- I Combinatorial Bounds.- 3 Johnson-Type Bounds and Applications to List Decoding.- 4 Limits to List Decodability.- 5 List Decodability Vs. Rate.- II Code Constructions and Algorithms.- 6 Reed-Solomon and Algebraic-Geometric Codes.- 7 A Unified Framework for List Decoding of Algebraic Codes.- 8 List Decoding of Concatenated Codes.- 9 New, Expander-Based List Decodable Codes.- 10 List Decoding from Erasures.- III Applications.- Interlude.- III Applications.- 11 Linear-Time Codes for Unique Decoding.- 12 Sample Applications Outside Coding Theory.- 13 Concluding Remarks.- A GMD Decoding of Concatenated Codes.