List Decoding of Error-Correcting Codes

Winning Thesis of the 2002 ACM Doctoral Dissertation Competition
2005
ISBN: 978-3-540-24051-8
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 3282, 352 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1160 g

Reihe: Lecture Notes in Computer Science

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.

Zielgruppe

Research

Autoren/Hrsg.

Guruswami, Venkatesan

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie
• Mathematik | Informatik EDV | Informatik Informatik Logik, formale Sprachen, Automaten
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Informationstheorie, Kodierungstheorie

Inhaltsverzeichnis

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.