Venkatesan Guruswami (auth.)3540240519, 9783540240518, 9783540301806
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. |
Table of contents :
Front Matter….Pages –
1 Introduction….Pages 1-14
2 Preliminaries and Monograph Structure….Pages 15-30
Front Matter….Pages 31-31
3 Johnson-Type Bounds and Applications to List Decoding….Pages 33-44
4 Limits to List Decodability….Pages 45-78
5 List Decodability Vs. Rate….Pages 79-92
Front Matter….Pages 93-93
6 Reed-Solomon and Algebraic-Geometric Codes….Pages 95-145
7 A Unified Framework for List Decoding of Algebraic Codes….Pages 147-175
8 List Decoding of Concatenated Codes….Pages 177-207
9 New, Expander-Based List Decodable Codes….Pages 209-250
10 List Decoding from Erasures….Pages 251-277
Front Matter….Pages 279-279
Interlude….Pages 281-281
11 Linear-Time Codes for Unique Decoding….Pages 283-298
12 Sample Applications Outside Coding Theory….Pages 299-327
13 Concluding Remarks….Pages 329-332
A GMD Decoding of Concatenated Codes….Pages 333-335
Back Matter….Pages – |
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