Bridging Shannon and Hamming: List Error-correction with Optimal Rate

Guruswami, Venkatesan

doi:10.1142/9789814324359_0162

Cited by 4 publications

(4 citation statements)

References 53 publications

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“…Results for non-adversarial error models. All the results mentioned above are for the adversarial error model of Hamming [Ham50,Gur10]. In the setting of random corruptions (Shannon model), the situation seems to be better understood thanks to the seminal result on explicit polar codes of Arikan [Ari09].…”

Section: Do Explicit Binary Codes Near the Gv Bound Admit An Efficien...mentioning

confidence: 99%

“…Binary error correcting codes have pervasive applications [Gur10,GRS19] and yet we are far from understanding some of their basic properties [Gur09]. For instance, until very recently no explicit binary code achieving distance 1/2 − ε/2 with rate near Ω(ε 2 ) was known, even though the existence of such codes was (non-constructively) established long ago [Gil52,Var57] in what is now referred as the Gilbert-Varshamov (GV) bound.…”

Section: Introductionmentioning

confidence: 99%

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Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

Jeronimo¹,

Quintana²,

Srivastava³

et al. 2020

Preprint

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The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance 1/2 − ε and rate Ω(ε 2 ) (where an upper bound of O(ε 2 log(1/ε)) is known). Ta-Shma [STOC 2017] gave an explicit construction of ε-balanced binary codes, where any two distinct codewords are at a distance between 1/2 − ε/2 and 1/2 + ε/2, achieving a near optimal rate of Ω(ε 2+β ), where β → 0 as ε → 0.We develop unique and list decoding algorithms for (a slight modification of) the family of codes constructed by Ta-Shma, in the adversarial error model. We prove the following results for ε-balanced codes with block length N and rate Ω(ε 2+β ) in this family:-For all ε, β > 0 there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time N O ε,β (1) .-For any fixed constant β independent of ε, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time (log(1/ε)) O(1) • N O β (1) .-For any ε > 0, there are explicit ε-balanced codes with rate Ω(ε 2+β ) which can be list 1) , where ε ′ , β → 0 as ε → 0.The starting point of our algorithms is the framework for list decoding direct-sum codes develop in Alev et al. [SODA 2020], which uses the Sum-of-Squares SDP hierarchy. The rates obtained there were quasipolynomial in ε. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of Ta-Shma's construction.

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Section: Do Explicit Binary Codes Near the Gv Bound Admit An Efficien...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

Jeronimo¹,

Quintana²,

Srivastava³

et al. 2020

Preprint

View full text Add to dashboard Cite

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“…The breakthrough works of Parvaresh-Vardy [36] and Guruswami-Rudra [23] gave families of codes which could be (efficiently) list decoded beyond the Johnson bound, and were followed by several related combinatorial and algorithmic results for other codes (e.g., [9,19,29,18]). For more detailed surveys on list decoding of codes we refer to [43,20,21,22].…”

Section: The Johnson Boundmentioning

confidence: 99%

List Decoding Barnes-Wall Lattices

Grigorescu

Peikert

2012

2012 IEEE 27th Conference on Computational Complexity

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The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in R N , and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L ⊆ R N , given a target vector r ∈ R N and a distance parameter d, output the set of all lattice points w ∈ L that are within distance d of r.In this work we focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes-Wall lattices. Our main contributions are twofold:1. We give tight (up to polynomials) combinatorial bounds on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattice's minimum distance (in the Euclidean norm).2. Building on the unique decoding algorithm of Micciancio and Nicolosi (ISIT '08), we give a listdecoding algorithm that runs in time polynomial in the lattice dimension and worst-case list size, for any error radius. Moreover, our algorithm is highly parallelizable, and with sufficiently many processors can run in parallel time only poly-logarithmic in the lattice dimension.In particular, our results imply a polynomial-time list-decoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.

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“…The broadcast channel with certain receiver side information under list decoding is studied in [25]. Recently, the concept of list decoding attracted attention also for error-correction codes [6,17,18].…”

Section: Introductionmentioning

confidence: 99%

Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources

Boche¹,

Schaefer²

2016

AMC

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Communication over channels that may vary in an arbitrary and unknown manner from channel use to channel use is studied. Such channels fall in the framework of arbitrarily varying channels (AVCs), for which it has been shown that the classical deterministic approaches with pre-specified encoder and decoder fail if the AVC is symmetrizable. However, more sophisticated strategies such as common randomness (CR) assisted codes or list decoding are capable to resolve the ambiguity induced by symmetrizable AVCs. AVCs further serve as the indispensable basis for modeling adversarial attacks such as jamming in information theoretic security related communication problems. In this paper, we study the arbitrarily varying multiple access channel (AVMAC) with conferencing encoders, which models the communication scenario with two cooperating transmitters and one receiver. This can be motivated for example by cooperating base stations or access points in future systems. The capacity region of the AVMAC with conferencing encoders is established and it is shown that list decoding allows for reliable communication also for symmetrizable AVMACs. The list capacity region equals the CR-assisted capacity region for large enough list size. Finally, for fixed probability of decoding error the amount of resources, i.e., CR or list size, is quantified and shown to be finite.

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Bridging Shannon and Hamming: List Error-correction with Optimal Rate

Cited by 4 publications

References 53 publications

Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

List Decoding Barnes-Wall Lattices

Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources

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