Combinatorial Construction of Locally Testable Codes

Meir, Or

doi:10.1137/080729967

Cited by 41 publications

(38 citation statements)

References 22 publications

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“…Moreover, in the area of LTCs we usually require δ(C)/3. E.g., all known constructions of LTCs satisfy this requirement (see e.g., [10], [12], [20], [23], [24], [30]). On the other side, if for all constants q, > 0 the code C is not (q, , δ(C)/3)-LTC we say that C is not locally testable (see e.g., [4], [8], [25]).…”

Section: A Ltcs and Ldcsmentioning

confidence: 96%

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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Section: A Ltcs and Ldcsmentioning

confidence: 96%

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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“…Moreover, in the area of LTCs we usually require δ(C )/3. For example, all known constructions of LTCs satisfy this requirement, see (Ben-Sasson & Sudan 2008;Dinur 2007;Kaufman & Sudan 2007Meir 2009)). On the other side, if for all constants q, > 0 the code C is not (q, , δ(C )/3)-LTC we say that C is not locally testable (Babai et al 2005;Kaufman & Viderman 2010).…”

Section: Ltcs and Ldcsmentioning

confidence: 99%

“…We point out all known constructions of LTCs seem to be dense and natural, although the best known LTCs (e.g. Ben-Sasson & Sudan 2008;Dinur 2007;Meir 2009) are not regular but still seem to be dense and natural. Conjecture 3.11 (3-query LTCs are natural and dense).…”

Section: Dense Natural and Regular Ltcs Cannot Be Asymptotically Goodmentioning

confidence: 99%

Towards lower bounds on locally testable codes via density arguments

Ben–Sasson

Viderman

2012

comput. complex.

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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, it is sufficient to prove that the number of dual codewords of weight at most 3 is super-linear in the blocklength of the code and they are distributed "naturally". The main technical contribution of this paper is an improvement of the combinatorial lemma of Goldreich et al. (Comput Complex 15(3):263-296, 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 non-trivial 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 (IEEE conference on computational complexity. IEEE Computer Society, pp 291-298, 2010) observation that sufficiently strong bounds on the rate of such LDCs imply explicit constructions of rigid matrices.

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“…We end this section by pointing out that recently, tensor codes have played a role in the combinatorial construction by Meir [13] of quasilinear length locally testable codes. Better base codes may result in LTCs with improved rate, hence the importance in broadening the class of base codes that can be used to construct robustly testable tensor codes.…”

Section: Test For R ⊗Cmentioning

confidence: 99%