On Matrix Rigidity and Locally Self-correctable Codes

Dvir, Zeev

doi:10.1007/s00037-011-0009-1

Cited by 19 publications

(20 citation statements)

References 22 publications

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“…, a d ), as Enc(ā) =ā·G. We also note that in several previous works (e.g., in [7]), LCCs are defined by means of their dual matrix but, for our purposes, this (equivalent) definition, in terms of the generating matrix, will be more convenient.…”

Section: The Main Theoremmentioning

confidence: 97%

“…For instance, one can hope that such understanding could lead to explicit constructions of rigid matrices [2,7]. An example of the usefulness of such bounds is demonstrated by the work of Alon [1], that proved lower bound on the ranks of perturbed identity matrices.…”

Section: A Rank Bound For Design Matrices Over Finite Fieldsmentioning

confidence: 99%

“…We can now apply Theorem 3.1 to find a subset A ⊆ A of size |A | ≥ (γ/2)|A| such that (7) |A − A | ≤ (4/γ) 8 |B| 4 /|A| 3 .…”

Section: Proof Of Lemma 44mentioning

confidence: 99%

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Tight lower bounds for linear 2-query LCCs over finite fields

et al. 2015

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A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a δ fraction of the coordinates is corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions.In this work, we show a separation between linear 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form p Ω(δd) on the length of linear 2-query LCCs over Fp, that encode messages of length d. Our bound improves over the known bound of 2 Ω(δd) [8,10,6] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science.We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields.The paper also contains an appendix, written by Sergey Yekhanin, showing that there do exist nonlinear LCCs of size 2 O(d) over Fp, thus highlighting the importance of the linearity assumption for our result.

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Section: The Main Theoremmentioning

confidence: 97%

Section: A Rank Bound For Design Matrices Over Finite Fieldsmentioning

confidence: 99%

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Tight lower bounds for linear 2-query LCCs over finite fields

et al. 2015

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“…Previous lower bounds on LDCs with δ = o(1) were not achieved because of lack of tight lower bounds on 2-query LDCs with very small but non-trivial δ, i.e., where ω(1) ≤ δn ≤ log n (see Dvir [15] for motivation for such bounds). In Theorem III.16 we give such a lower bound.…”

Section: B Limiting the Rate Of Weak 2-query Ldcsmentioning

confidence: 98%

“…First, given that state-of-theart bounds on rate of q-query LDCs for q ≥ 3 rely on rate bounds 2-query LDCs with a sublinear number of errors [34], [35] shows that proving rate bounds for smaller values should result in improved bounds for qquery LDCs even for larger values of q. Second, the recent work of Dvir [15] shows that proving sufficiently strong lower bounds on locally decodable codes which can be corrected from a sublinear number of corruptions would result in explicit constructions of rigid matrices, giving further motivation for our lemma.…”

Section: Introductionmentioning

confidence: 99%

Towards Lower Bounds on Locally Testable Codes via Density Arguments

Ben–Sasson

Viderman

2011

2011 IEEE 26th Annual Conference on Computational Complexity

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The main open problem in the area of locally testable codes (LTCs) is whether there exists an asymptotically good family of LTCs and to resolve this question it suffices to consider the case of query complexity 3. We argue that to refute the existence of such an asymptotically good family one should prove that the number of dual codewords of weight at most 3 is super-linear in the blocklength of the code.The main technical contribution of this paper is an improvement of the combinatorial lemma of Goldreich et al. [2006] which bounds the rate of 2-query locally decodable codes (LDCs) and is used in state-of-the-art rate-bounds for linear LDCs. The lemma of Goldreich et al. bounds the rate of 2-query LDCs of blocklength n in terms of the corruption parameter δ(n) -this is the maximal fraction of corrupted codeword bits for which a (2-query) decoder can recover correctly every message bit (with high probability). Our combinatorial lemma gives nontrivial rate bounds for any corruption parameter δ(n) such that δ(n) · n = ω(1), whereas the previous lemma works only for corruption parameter δ(n) such that δ(n) · n ≥ log n. The study of LDCs with sublinear corruption parameter is also motivated by Dvir's [2010] observation that sufficiently strong bounds on the rate of such LDCs imply explicit constructions of rigid matrices.

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A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field

Woodruff

2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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A linear (q, δ, , m(n))-locally decodable code (LDC) C : F n → F m(n) is a linear transformation from the vector space F n to the space F m(n) for which each message symbol xi can be recovered with probability at least 1 |F| + from C(x) by a randomized algorithm that queries only q positions of C(x), even if up to δm(n) positions of C(x) are corrupted. In a recent work of Dvir, the author shows that lower bounds for linear LDCs can imply lower bounds for arithmetic circuits. He suggests that proving lower bounds for LDCs over the complex or real field is a good starting point for approaching one of his conjectures. Our main result is an m(n) = Ω(n 2) lower bound for linear 3-query LDCs over any, possibly infinite, field. The constant in the Ω(·) depends only on ε and δ. This is the first lower bound better than the trivial m(n) = Ω(n) for arbitrary fields and more than two queries.

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On Matrix Rigidity and Locally Self-correctable Codes

Cited by 19 publications

References 22 publications

Tight lower bounds for linear 2-query LCCs over finite fields

Tight lower bounds for linear 2-query LCCs over finite fields

Towards Lower Bounds on Locally Testable Codes via Density Arguments

A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field

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