The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound

Thommesen, Christian

doi:10.1109/tit.1983.1056765

Cited by 31 publications

(36 citation statements)

References 2 publications

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“…Overview of our construction. Our randomized construction is based on an ensemble of codes that was considered by Thommesen to construct randomized binary codes that lie on the Gilbert-Varshamov bound [30]. C out is the Reed-Solomon code but the inner codes are chosen to be independent random binary codes.…”

Section: Techniquesmentioning

confidence: 99%

“…C out is the Reed-Solomon code but the inner codes are chosen to be independent random binary codes. Unlike [30], where the inner codes are linear, in our case the (non-linear) inner codes are picked as follows: every codeword is chosen to be a random vector in {0, 1} n2 , where each entry is chosen to be 1 with probability Θ(1/d). We show that this concatenated code with high probability gives rise to a d-disjunct matrix.…”

Section: Techniquesmentioning

confidence: 99%

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Efficiently Decodable Non-adaptive Group Testing

Indyk¹,

Ngo²,

Rudra³

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

118

127

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We consider the following "efficiently decodable" nonadaptive group testing problem. There is an unknown string x ∈ {0, 1} n with at most d ones in it. We are allowed to test any subset S ⊆ [n] of the indices. The answer to the test tells whether xi = 0 for all i ∈ S or not. The objective is to design as few tests as possible (say, t tests) such that x can be identified as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics.A non-adaptive group testing strategy can be represented by a t × n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d 2 log n) efficiently. However, these matrices so far only allow for a "decoding" time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d.This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d 2 log n) that can be decoded in time poly(d) · t log 2 t + O(t 2 ). To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d 2 log n) bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n/ log log n).A crucial building block in our construction is the notion of (d, )-list disjunct matrices, which represent the more general "list group testing" problem whose goal is to output less than d + positions in x, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices, expanders, dispersers and disjunct matrices. List disjunct matrices have applications in constructing (d, )-sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers for Reed-Solomon codes in the data stream model.

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Section: Techniquesmentioning

confidence: 99%

Section: Techniquesmentioning

confidence: 99%

Efficiently Decodable Non-adaptive Group Testing

Indyk¹,

Ngo²,

Rudra³

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

118

127

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“…The starting point for our constructions is the random concatenation technique of Thommesen [Tho83], which he used to show that codes of a particular simple form can achieve the GV bound. Specifically, he showed that if one takes a ReedSolomon code over a large alphabet as the outer code, and concatenate it with binary linear inner codes chosen uniformly at random and independently for each outer coordinate, then the resulting code C lies on the GV bound with high probability.…”

Section: Methodsmentioning

confidence: 99%

“…Thommesen [Tho83] used the operation of random concatenation to construct a binary code lying on the GV bound out of a large alphabet code lying on the Singleton bound. The following lemma shows an approximate version of this, replacing "lying on" with "close to".…”

Section: Approaching the Gv Bound Via Random Concatenationmentioning

confidence: 99%

Locally Testable and Locally Correctable Codes Approaching the Gilbert-Varshamov Bound

Gopi

Kopparty

Oliveira

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every δ ∈ (0, 1/2), there are codes with minimum distance δ and rate R = R GV (δ) > 0 (for a certain simple function R GV (·)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and errorcorrection algorithms. Specifically, we show the following results.1. Local Testing: For all δ ∈ (0, 1/2) and all R < R GV (δ), there exist codes with length n, rate R and minimum distance δ that are locally testable with quasipolylog(n) query complexity. Local Correction:For all > 0, for all δ < 1/2 sufficiently large, and all R < (1 − )R GV (δ), there exist codes with length n, rate R and minimum distance δ that are locally correctable from Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters.Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, AlonEdmonds-Luby distance amplification, and the local listdecodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.

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“…We will need the following property of this function, which was proven in [16] for the q = 2 case. The following is an easy extension of the result for general q.…”

mentioning

confidence: 99%

The Existence of Concatenated Codes List-Decodable up to the Hamming Bound

Guruswami

Rudra

2010

IEEE Trans. Inform. Theory

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We prove that binary linear concatenated codes with an outer algebraic code (specifically, a folded Reed-Solomon code) and independently and randomly chosen linear inner codes achieve, with high probability, the optimal trade-off between rate and list-decoding radius. In particular, for any 0 < ρ < 1/2 and ε > 0, there exist concatenated codes of rate at least 1 − H(ρ) − ε that are (combinatorially) list-decodable up to a ρ fraction of errors. (The Hamming bound states that the best possible rate for such codes cannot exceed 1 − H(ρ), and standard random coding arguments show that this bound is approached by random codes with high probability.) A similar result, with better list size guarantees, holds when the outer code is also randomly chosen. Our methods and results extend to the case when the alphabet size is any fixed prime power q 2.Our result shows that despite the structural restriction imposed by code concatenation, the family of concatenated codes is rich enough to include codes list-decodable up to the optimal Hamming bound. This provides some encouraging news for tackling the problem of constructing explicit binary list-decodable codes with optimal rate, since code concatenation has been the preeminent method for constructing good codes for list decoding over small alphabets.

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The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound

Cited by 31 publications

References 2 publications

Efficiently Decodable Non-adaptive Group Testing

Efficiently Decodable Non-adaptive Group Testing

Locally Testable and Locally Correctable Codes Approaching the Gilbert-Varshamov Bound

The Existence of Concatenated Codes List-Decodable up to the Hamming Bound

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