Error-correcting codes are combinatorial objects designed to cope with the problem of reliable transmission of information on a noisy channel. A fundamental algorithmic challenge in coding theory and practice is to efficiently decode the original transmitted message even when a few symbols of the received word are in error. The naive search algorithm runs in exponential time, and several classical polynomial time decoding algorithms are known for specific code families. Traditionally, however, these algorithms have been constrained to output a unique codeword. Thus they faced a “combinatorial barrier” and could only correct up to d/2 errors, where d is the minimum distance of the code. 
An alternate notion of decoding called list decoding, proposed independently by Elias and Wozencraft in the late 50s, allows the decoder to output a list of all codewords that differ from the received word in a certain number of positions. Even when constrained to output a relatively small number of answers, list decoding permits recovery from errors well beyond the d/2 barrier, and opens up the possibility of meaningful error-correction from large amounts of noise. However, for nearly four decades after its conception, this potential: of list decoding was largely untapped due to the lack of efficient algorithms to list decode beyond d/2 errors for useful families of codes. 
This thesis presents a detailed investigation of list decoding, and proves its potential, feasibility, and importance as a combinatorial and algorithmic concept. We prove several; combinatorial results that sharpen our understanding of the potential and limits of list; decoding, and its relation to more classical parameters like the rate and minimum distance. The crux of the thesis is its algorithmic results, which were lacking in the early works on list decoding. Our algorithmic results include: (1) Efficient list decoding algorithms for classically studied codes such as Reed-Solomon codes and algebraic-geometric codes. In particular, building upon an earlier algorithm due to Sudan, we present the first polynomial time algorithm to decode Reed-Solomon codes beyond d/2 errors for every value of the rate. (2) A new soft list decoding algorithm for Reed-Solomon and algebraic-geometric codes and novel decoding algorithms for concatenated codes based on it. (3) New code constructions using concatenation and/or expander graphs that have good (and sometimes near-optimal) rate and are efficiently list decodable from extremely large amounts of noise. (4) Expander-based constructions of linear time encodable and decodable codes that ca4 correct up to the maximum possible fraction of errors, using unique (not list) decoding. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

List Decoding of Error-Correcting Codes

Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of “list-decoding” for error-correcting codes. In this survey we describe the list-decoding problem, the algorithms that have been developed, and a diverse collection of applications within complexity theory.

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List decoding: algorithms and applications

We present several novel constructions of codes which share the common thread of using expander (or expander-like) graphs as a component. The expanders enable the design of efficient decoding algorithms that correct a large number of errors through various forms of "voting" procedures. We consider both the notions of unique and list decoding, and in all cases obtain asymptotically good codes which are decodable up to a "maximum" possible radius and either: (a) achieve a similar rate as the previously best known codes but come with significantly faster algorithms, or (b) achieve a rate better than any prior construction with similar error-correction properties. Among our main results are: i) codes of rate /spl Omega/(/spl epsi//sup 2/) over constant-sized alphabet that can be list decoded in quadratic time from (1-/spl epsi/) errors; ii) codes of rate /spl Omega/(/spl epsi/) over constant-sized alphabet that can be uniquely decoded from (1/2-/spl epsi/) errors in near-linear time (this matches AG-codes with much faster algorithms); iii) linear-time encodable and decodable binary codes of positive rate (in fact, rate /spl Omega/(/spl epsi//sup 2/)) that can correct up to (1/4-/spl epsi/) fraction errors.

Expander-based constructions of efficiently decodable codes

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a fraction (1-r-/spl epsiv/)/2 of errors over an alphabet of constant size depending only on /spl epsiv/, for every 0 0. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are "near-MDS" codes). Such near-MDS linear-time codes were known for the decoding from erasures; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the Zyablov bound, and also the more general Blokh-Zyablov bound (by resorting to multilevel concatenation). Our work also yields linear-time encodable/decodable codes which match Forney's error exponent for concatenated codes for communication over the binary symmetric channel. The encoding/decoding complexity was quadratic in Forney's result, and Forney's bound has remained the best constructive error exponent for almost 40 years now. In summary, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in addition have linear-time encoding and decoding algorithms.

Linear-time encodable/decodable codes with near-optimal rate

Informally, an error-correcting code has "nice" list-decodability properties if every Hamming ball of "large" radius has a "small" number of codewords in it. We report linear codes with nontrivial list-decodability: i.e., codes of large rate that are nicely list-decodable, and codes of large distance that are not nicely list-decodable. Specifically, on the positive side, we show that there exist codes of rate R and block length n that have at most c codewords in every Hamming ball of radius H/sup -1/(1-R-1/c)/spl middot/n. This answers the main open question from the work of Elias (1957). This result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors, improving previous results of Guruswami and Sudan (see IEEE Trans. Inform. Theory, vol.45, p.1757-67, Sept. 1999, and Proc. 32nd ACM Symp. Theory of Computing (STOC), Portland, OR, p. 181-190, May 2000) in this vein. Specifically, for every /spl epsi/ > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate /spl Omega/(/spl epsi//sup 4/) that can be list-decoded in polynomial time from up to a fraction (1/2-/spl epsi/) of errors, using lists of size O(/spl epsi//sup -2/). On the negative side, we show that for every /spl delta/ and c, there exists /spl tau/ 0, and an infinite family of linear codes {C/sub i/}/sub i/ such that if n/sub i/ denotes the block length of C/sub i/, then C/sub i/ has minimum distance at least /spl delta/ /spl middot/ n/sub i/ and contains more than c/sub 1/ /spl middot/ n/sub i//sup c/ codewords in some Hamming ball of radius /spl tau/ /spl middot/ n/sub i/. While this result is still far from known bounds on the list-decodability of linear codes, it is the first to bound the "radius for list-decodability by a polynomial-sized list" away from the minimum distance of the code.

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Combinatorial bounds for list decoding

We generalize Sudan's (see J. Compl., vol.13, p.180-93, 1997) results for Reed-Solomon codes to the class of algebraic-geometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error-correction bound (d-1)/2, d being the minimum distance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomial-time algorithm and show that the resulting overall list-decoding algorithm runs in polynomial time under some mild conditions. Several examples are included.

H. Wasserman

Papers

List decoding of algebraic-geometric codes