High-rate codes with sublinear-time decoding

Kopparty, Swastik; Saraf, Shubhangi; Yekhanin, Sergey

doi:10.1145/1993636.1993660

Cited by 77 publications

(142 citation statements)

References 54 publications

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“…Definition 3 (Multiplicity Codes [3]): Given the above evaluation map and a degree d < sq, the corresponding Multiplicity code is…”

Section: M+s−1 Mmentioning

confidence: 99%

“…For these codes, the message space and the codeword space do not share the same alphabet, so the standard reduction from [2, Lemma 2.3] can not be applied. The problem was circumvented in [3] by using concatenation.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], for the very particular case of Reed-Muller codes and Multiplicity codes, a better usage of these locally decodable codes in PIR schemes was introduced, using a partitioning of the m-dimensional affine space into few affine hyperplanes. The concatenation Ngô is supported by INRIA and Alcatel-Lucent Privacy ADR solution provided by [3] appears not helpful in this context, since it more or less breaks the underlying affine geometry.…”

Section: Introductionmentioning

confidence: 99%

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Information sets of Multiplicity codes

Augot

Levy-dit-Vehel

Ngo

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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Abstract-We here provide a method for systematic encoding of the Multiplicity codes introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built on an idea of Kopparty. We properly define information sets for these codes and give detailed proofs of the validity of Kopparty's construction, that use generating functions. We also give a complexity estimate of the associated encoding algorithm.

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“…Definition 3 (Multiplicity Codes [3]): Given the above evaluation map and a degree d < sq, the corresponding Multiplicity code is…”

Section: M+s−1 Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Information sets of Multiplicity codes

Augot

Levy-dit-Vehel

Ngo

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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“…Katz and Trevisan [239] introduced the notion of smooth codes and showed their close relation to both private information retrieval and locally decodable codes (Problem 7.6). Recently, Saraf, Kopparty, and Yekhanin [249] constructed the first locally decodable codes with sublinear-time decoding and rate larger 1/2.…”

Section: Chapter Notes and Referencesmentioning

confidence: 99%

Chapter 8 Notes and References

2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

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“…Thus, it is natural to ask whether we can get PCPs with better length if we allow ourselves to use more queries, say, O(n ε ) queries. Indeed, recent constructions of locally decodable codes [KSY11], and locally testable codes [Vid12], show that by allowing O(n ε ) queries, it is possible to achieve very high rates, arbitrarily close to 1.…”

Section: Introductionmentioning

confidence: 99%

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Ben–Sasson

Kaplan

Kopparty

et al. 2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n ε , then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(t log t) for any language in NTIME(t).More specifically, for any ε > 0 we present (non-uniform) probabilistically checkable proofs (PCPs) of length 2 O(1/ε) · n that can be checked using n ε queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this paper for the first time for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth [Trans. Information Theory 2006].

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High-rate codes with sublinear-time decoding

Cited by 77 publications

References 54 publications

Information sets of Multiplicity codes

Information sets of Multiplicity codes

Chapter 8 Notes and References

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

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