Construction Of Asymptotically Good Low-rate Error-correcting Codes Through Pseudo-random Graphs

Alon, N.; Bruck, Jehoshua; Naor, Joseph; Naor, Moni; Roth, Ron M.

doi:10.1109/isit.1991.695195

Cited by 60 publications

(150 citation statements)

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“…The area fell dormant for quite a few years, and coding theorists concentrated on algebraic techniques, until graph codes stormed again into consciousness in the 80's and 90's, with important works of Tanner [Tan81], Alon et al [ABN92], Sipser and Spielman [SS96], and more. This particular direction culminated in the paper of Spielman [Spi96], which achieved linear time encoding and decoding for asymptotically good codes.…”

Section: Error Correcting Codesmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Error Correcting Codesmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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“…A very similar function is studied in [6], and most of the techniques applied there can be used in our case as well, as we briefly describe below. Besides these techniques, we need a new result, stated in proposition 4 below.…”

Section: The Smallest Possible -Bias Spacesmentioning

confidence: 99%

“…A lower bound for m(n, ) can be derived-(as is also mentioned in [6] for the case of fixed )-from the McEliece-Rodemich-Rumsey-Welch bound (see [23], page 559). Although the proof of this bound, as described, e.g., in [23] is given only for the case of a fixed (when the length of the code tends to infinity), the same proof can be extended to a more general case, by studying the asymptotic behavior of the smallest roots of the corresponding Krawtchouk polynomials.…”

Section: The Smallest Possible -Bias Spacesmentioning

confidence: 99%

Simple Constructions of Almost k‐wise Independent Random Variables

Alon

Goldreich

Håstad

et al. 1992

Random Struct Algorithms

446

358

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We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ), where is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as < 1/(k log n)). An additional advantage of our constructions is their simplicity.

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“…The way Hamming spheres pack in high-dimensional space, even for p much larger than (1 − R)/2 (and in fact for p ≈ 1 − R) there exist codes of rate R (over a larger alphabet Σ) for which the following holds: for most error patterns e that corrupt fewer than a fraction p of symbols, when a codeword c gets distorted into z by the error pattern e, there will be no codeword besides c within Hamming distance pn of z. 1 Thus, for typical noise patterns one can hope to correct many more errors than the above limit faced by the worst-case error pattern. However, since we assume a worst-case noise model, we do have to deal with bad received words such as r. List decoding provides an elegant formulation to deal with worst-case errors without compromising the performance for typical noise patterns -the idea is that in the worst-case, the decoder may output multiple answers.…”

Section: List Decoding: Context and Motivation 111mentioning

confidence: 99%

Algorithmic Results in List Decoding

Guruswami

2006

FNT in Theoretical Computer Science

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Error-correcting codes are used to cope with the corruption of data by noise during communication or storage. A code uses an encoding procedure that judiciously introduces redundancy into the data to produce an associated codeword. The redundancy built into the codewords enables one to decode the original data even from a somewhat distorted version of the codeword. The central trade-off in coding theory is the one between the data rate (amount of non-redundant information per bit of codeword) and the error rate (the fraction of symbols that could be corrupted while still enabling data recovery). The traditional decoding algorithms did as badly at correcting any error pattern as they would do for the worst possible error pattern. This severely limited the maximum fraction of errors those algorithms could tolerate. In turn, this was the source of a big hiatus between the error-correction performance known for probabilistic noise models (pioneered by Shannon) and what was thought to be the limit for the more powerful, worst-case noise models (suggested by Hamming).In the last decade or so, there has been much algorithmic progress in coding theory that has bridged this gap (and in fact nearly eliminated it for codes over large alphabets). These developments rely on an error-recovery model called "list decoding," wherein for the pathological error patterns, the decoder is permitted to output a small list of candidates that will include the original message. This book introduces and motivates the problem of list decoding, and discusses the central algorithmic results of the subject, culminating with the recent results on achieving "list decoding capacity." Part IGeneral Literature

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Construction Of Asymptotically Good Low-rate Error-correcting Codes Through Pseudo-random Graphs

Cited by 60 publications

References 0 publications

Expander graphs and their applications

Expander graphs and their applications

Simple Constructions of Almost k‐wise Independent Random Variables

Algorithmic Results in List Decoding

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