High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity

Kopparty, Swastik; Meir, Or; Ron-Zewi, Noga; Saraf, Shubhangi

doi:10.1145/2897518.2897523

Cited by 27 publications

(37 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, this resiliency to the worst-case error leads to coding limitations and some possibly undesirable tradeoffs. On one hand, current constructions for LDCs that focus on efficient encoding can obtain any constant rate R < 1 while simultaneously being robust to any constant fraction δ < 1−R of errors and using O 2 √ log n log log n queries for decoding [KMRS17]. On the other hand, codes that focus on low query complexity obtain blocklength that is subexponential in the message length while using a constant number of queries q ≤ 3 [Yek08, Efr12,DGY11].…”

Section: Related Workmentioning

confidence: 99%

Relaxed Locally Correctable Codes in Computationally Bounded Channels

Blocki¹,

Gandikota

Grigorescu³

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

Constructions of locally decodable codes (LDCs) have one of two undesirable properties: low rate or high locality (polynomial in the length of the message). In settings where the encoder/decoder have already exchanged cryptographic keys and the channel is a probabilistic polynomial time (PPT) algorithm, it is possible to circumvent these barriers and design LDCs with constant rate and small locality. However, the assumption that the encoder/decoder have exchanged cryptographic keys is often prohibitive. We thus consider the problem of designing explicit and efficient LDCs in settings where the channel is slightly more constrained than the encoder/decoder with respect to some resource e.g., space or (sequential) time.Given an explicit function f that the channel cannot compute, we show how the encoder can transmit a random secret key to the local decoder using f (·) and a random oracle H(·). This allows bootstrap from the private key LDC construction of Ostrovsky, Pandey and Sahai (ICALP, 2007), thereby answering an open question posed by Guruswami and Smith (FOCS 2010) of whether such bootstrapping techniques may apply to LDCs in weaker channel models than just PPT algorithms. Specifically, in the random oracle model we show how to construct explicit constant rate LDCs with optimal locality of polylog in the security parameter against various resource constrained channels. √ log n log log n [KMRS17]. Our codes are robust against constant fraction of corruptions, and are non-adaptive i.e. all the queries made by the decoder are independent of each other, a property which may be desired when it is beneficial to submit all local decoding queries in a single batch.Our constructions stand at the intersection of coding theory and cryptography, using well-known tools and techniques from cryptography to provide notions of (information theoretic) randomness and security for communication protocols between sender/receiver. To prove the security of our constructions, we introduce a two-phase distinguisher hybrid argument, which may be of independent interest to apply to other coding theoretic problems in these resource bounded channel models. Preliminaries NotationWe use the notation [n] to represent the set {1, 2, . . . , n}. For any x, y ∈ Σ n , let HAM(x) denote the Hamming weight of x, i.e. the number of non-zero coordinates of x. Let HAM(x, y) = HAM(x − y) 1 Note that small alphabet sizes are attractive for practical channels designed to transmit bits efficiently. 2 In this paper we use the security parameter κ in an asymptotic sense e.g., for any attacker running in time poly(κ) there is a negligible function negl(κ) upper bounding the probability that the attacker succeeds. In particular, the function negl(κ) = 2 − log 1+ε κ) is negligible, but does not provide κ-bits of concrete security i.e., any attacker running in time t succeeds with probability at most t2 −κ .3 [OPS07] gives a construction with locality f (κ) = ω(log κ) "one-time" private LDCs, but the construction needs to be modified if we want security again...

show abstract

Section: Related Workmentioning

confidence: 99%

Relaxed Locally Correctable Codes in Computationally Bounded Channels

Blocki¹,

Gandikota

Grigorescu³

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

show abstract

“…It is known that the double-cover of a Ramanujan graph has the properties we want; we state these properties formally in the following claim. KMRS17], Lemma 2.7] Let ξ, ε, R ∈ [0, 1], so that ξ and ε are sufficiently small. For infinitely many integers N > 0, there exists a D = O(1/ξε 2 ) so that the following holds.…”

Section: A a Wronskian Lemmamentioning

confidence: 99%

Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

Kopparty

Ron-Zewi

Saraf³

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions of codes known to achieve list-decoding capacity; multivariate multiplicity codes were the first constructions of high-rate locally correctable codes; and univariate multiplicity codes are also known to achieve list-decoding capacity.However, previous analyses of the error-correction properties of these codes did not yield optimal results. In particular, in the list-decoding setting, the guarantees on the list-sizes were polynomial in the block length, rather than constant; and for multivariate multiplicity codes, local list-decoding algorithms could not go beyond the Johnson bound.In this paper, we show that Folded Reed-Solomon codes and multiplicity codes are in fact better than previously known in the context of list-decoding and local list-decoding. More precisely, we first show that Folded RS codes achieve list-decoding capacity with constant list sizes, independent of the block length; and that high-rate univariate multiplicity codes can also be list-recovered with constant list sizes. Using our result on univariate multiplicity codes, we show that multivariate multiplicity codes are high-rate, locally list-recoverable codes. Finally, we show how to combine the above results with standard tools to obtain capacity achieving locally list decodable codes with query complexity significantly lower than was known before. The rate of a code C ∈ Σ n is defined as R = 1 n log |Σ| (|C|) and quantifies how much information can be sent using the code. We always have R ∈ (0, 1), and we would like R to be as close to 1 as possible.3 They were previously shown to be list-decodable up to the Johnson bound by Nielsen [Nie01]. 4 Many codes in this paper have alphabet Σ = F s q , where Fq is a finite field. For such "vector alphabet" codes, we use the term "linear" to mean "Fq-linear".

show abstract

“…Capacity achieving locally list decodable codes. As mentioned above, one reason to seek high-rate codes is because of a transformation of Alon, Edmunds, and Luby [AEL95], recently highlighted in [KMRS16], which can, morally speaking, turn any high-rate code with a given property into a capacity achieving code with the same property. 1 This allows us to obtain capacity achieving locally list-decodable (or more generally, locally list recoverable) codes.…”

Section: Related Workmentioning

confidence: 99%

“…To get around this, we first encode our message with a high-rate locally decodable code, before encoding it with the tensor code. For this, we use the codes of [KMRS16], which have rate that is arbitrarily close to 1, and which are locally decodable with exp( √ log n) queries. This way, instead of directly querying the tensor code (which may give the wrong answer a constant fraction of the time), we instead use the outer locally decodable code to query the tensor code: this still does not use too many queries, but now it is robust to a few errors.…”

Section: Overview Of Techniquesmentioning

confidence: 99%

“…Let C : F ((1−ε/2)n) 1/t 2 s → F n 1/t 2 s be a linear code of rate (1−ε/2) 1/t ≤ 1−ε/(8t) (the inequality holds since (1 − x) y ≤ 1 − xy/4 for x, y ∈ [0, 1], see e.g. Fact 2.1 in [KMRS15]) that is ( ε 16t , ℓ, 2 O(sℓt/ε) )-list recoverable whose existence is guaranteed by Corollary 4.4 for sufficiently large n (depending on ε, ℓ, t, s). Note furthermore that by Theorem 4.1 we may assume that C has relative distance at least ε/(16t).…”

Section: Theorem 13 In [Kmrs15]mentioning

confidence: 99%

See 1 more Smart Citation

Local List Recovery of High-Rate Tensor Codes & Applications

Hemenway

Ron-Zewi

Wootters

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

In this work, we give the first construction of high-rate locally list-recoverable codes. Listrecovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known.Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are 'approximately' locally list-recoverable, and that the 'approximately' modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.

show abstract

High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity

Cited by 27 publications

References 55 publications

Relaxed Locally Correctable Codes in Computationally Bounded Channels

Relaxed Locally Correctable Codes in Computationally Bounded Channels

Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

Local List Recovery of High-Rate Tensor Codes & Applications

Contact Info

Product

Resources

About