3-query locally decodable codes of subexponential length

Efremenko, Klim

doi:10.1145/1536414.1536422

Cited by 165 publications

(189 citation statements)

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“…In a breakthrough work of Yekhanin [Yek08] and following improvements [Efr09, Rag07, KY09, IS10, CFL + 10, DGY11, BET10], a new family of LDCs based on Matching Vector families was introduced. These codes, called Matching-Vector codes, rely on constructions of MV families and can have sub-exponential length for q as small as 3 [Efr09]. Using Grolmuzs construction as a building block, one obtains an encoding length of roughly N ∼ exp exp (log K) O(log log q/ log q) (log log K) .…”

Section: Families and Locally Decodable Codesmentioning

confidence: 99%

Matching-Vector Families and LDCs over Large Modulo

Dvir

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We prove new upper bounds on the size of families of vectors in Z n m with restricted modular inner products, when m is a large integer. More formally, if u 1 , . . . , u t ∈ Z n m and v 1 , . . . , 2+8.47 ). This improves a recent bound of t ≤ m n/2+O(log(m)) by [BDL13] and is the best possible up to the constant 8.47 when m is sufficiently larger than n. The maximal size of such families, called 'Matching-Vector families', shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least K 19/18 for K-bit messages, regardless of their query complexity. This improves a known super linear bound of K2 Ω(

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Section: Families and Locally Decodable Codesmentioning

confidence: 99%

Matching-Vector Families and LDCs over Large Modulo

Dvir

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…Namely, Woodruff [2007] showed n must be at least (roughly) k 1+ 2 r whereas the best upper bounds of Yekhanin [2008] show n is at most (roughly) exp(k 1/ log log k ) (cf. Efremenko [2009]). …”

Section: Locally Correctable and Locally Testable Codes Affine-invarmentioning

confidence: 99%

“…In extreme cases where there is no abundance of local constraints, such as for low-density-parity-check (LDPC) codes based on random expanders, or for codes that have the very minimal number of local constraints needed to characterize them, there cannot be any hope for local testability , 2009. But, all things considered, abundance of local constraints should reduce the rate of the code, unless the constraints are carefully chosen in an algebraically consistent way.…”

Section: Locally Correctable and Locally Testable Codes Affine-invarmentioning

confidence: 99%

“…The existing work on affine invariant testing , Grigorescu et al, 2008, 2009 already converts questions about local testability of such codes into questions about solutions of certain systems of multivariate polynomial equations. (A solution of a system of equations is a common root, i.e., an assignment that sets all polynomials in the system to 0.)…”

Section: Zeroes Of Some Special Polynomial Systemsmentioning

confidence: 99%

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Limits on the Rate of Locally Testable Affine-Invariant Codes

Ben–Sasson¹,

Sudan²

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affine-invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be affine-invariant) could be found by systematically exploring the space of affine-invariant properties.In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on affine-invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by affine-invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.

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“…For example, the recent improved construction [16] of 3-query locally decodable codes of Yekhanin [41] relies crucially on a set system construction [21] which holds only modulo composite numbers. Generalizing results in the binary field (or prime fields) to commutative rings often poses new technical challenges and requires additional new ideas.…”

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confidence: 99%

Testing Non-uniform k-Wise Independent Distributions over Product Spaces

Rubinfeld

Xie

2010

Automata, Languages and Programming

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Abstract. A distribution D over Σ1 × · · · × Σn is called (non-uniform) k-wise independent if for any set of k indices {i1, . . . , i k } and for any z1We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al.[1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.A full version of this paper is available at

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3-query locally decodable codes of subexponential length

Cited by 165 publications

References 13 publications

Matching-Vector Families and LDCs over Large Modulo

Matching-Vector Families and LDCs over Large Modulo

Limits on the Rate of Locally Testable Affine-Invariant Codes

Testing Non-uniform k-Wise Independent Distributions over Product Spaces

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