Strong LTCs with Inverse Polylogarithmic Rate and Soundness

Viderman, Michael

doi:10.1109/ccc.2013.34

Cited by 4 publications

(23 citation statements)

References 17 publications

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“…Later, other constructions of LTCs [11,18,36] succeeded to obtain the rate 1 polylog(n) together with constant query complexity and soundness, however these codes were weak LTCs. It can be verified that every strong LTC is also a weak LTC, but some weak LTCs are not strong LTCs [45]. So, strong LTCs are strictly stronger objects than weak LTCs.…”

Section: Locally Testable Codesmentioning

confidence: 97%

“…In the journal version of [27], the authors pointed out that all known LTCs that achieve inverse polylogarithmic rate are weak LTCs, and asked about the existence of strong LTCs with polylogarithmic rate and, in particular, about the explicit construction of such codes [27,Section 6]. The previous papers of the author [45,44] showed a probabilistic construction of binary linear 3-query strong LTCs with inverse polylogarithmic rate, constant soundness and constant relative distance. In this paper (Section 1.4), we show the explicit construction of linear strong LTCs with constant query complexity, constant soundness, polylogarithmic rate and constant relative distance over a fixed field, therefore resolving a question raised by Goldreich and Sudan [27].…”

Section: Locally Testable Codesmentioning

confidence: 99%

“…As was mentioned previously, we prove that the codes of [11,18] yield explicit strong LTCs. To do that we want to reuse the arguments of [45,44] to prove that the codes of [11,18] are explicit strong LTCs. These codes (as well as codes of [36,45,44]) involve two kind of symbols: code symbols and proof symbols (called also core symbols and non-core symbols, respectively, in [45]).…”

Section: Locally Testable Codesmentioning

confidence: 99%

“…To do that we want to reuse the arguments of [45,44] to prove that the codes of [11,18] are explicit strong LTCs. These codes (as well as codes of [36,45,44]) involve two kind of symbols: code symbols and proof symbols (called also core symbols and non-core symbols, respectively, in [45]). 4 However, the codes of [11] have good distance only on the code symbols with no guarantee on the proof symbols, while the arguments used in [45] require both good distance on the code coordinates and on the proof coordinates.…”

Section: Locally Testable Codesmentioning

confidence: 99%

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Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

Viderman

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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An error-correcting code C ⊆ F n is called (q, )-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords x / ∈ C with probability at least · δ(x, C), where δ(x, C) denotes the relative Hamming distance between the word x and the code C. The parameter q is called the query complexity and the parameter is called soundness.Goldreich and Sudan (J.ACM 2006) asked about the existence of strong LTCs with constant query complexity, constant relative distance, constant soundness and inverse polylogarithmic rate. They also asked about the explicit constructions of these codes.Strong LTCs with the required range of parameters were obtained recently in the works of Viderman (CCC 2013, FOCS 2013) based on the papers of Meir (SICOMP 2009) and Dinur (J.ACM 2007). However, the construction of these codes was probabilistic.In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. Previously, such codes were proven to be weak LTCs. Using the results of Viderman (CCC 2013, FOCS 2013 we prove that such codes are, in fact, strong LTCs.

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Section: Locally Testable Codesmentioning

confidence: 97%

Section: Locally Testable Codesmentioning

confidence: 99%

Section: Locally Testable Codesmentioning

confidence: 99%

Section: Locally Testable Codesmentioning

confidence: 99%

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Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

Viderman

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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“…Remark 2.8. It is plausible that Definition 2.5 can be extended to contain also best known LTCs [16,23,39,47]. An interesting point is that the coordinates in these LTCs are partitioned into 2 subsets.…”

Section: Theorem 27 (Main Theorem) a Family F Of Linear Codes Is Smmentioning

confidence: 99%

A combinatorial characterization of smooth LTCs and applications

Ben–Sasson

Viderman

2016

Random Struct. Alg.

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The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet, we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs and whether asymptotically good linear LTCs with constant query complexity exist.In this paper, we provide a combinatorial characterization of smooth locally testable codes, which are locally testable codes whose associated tester queries every bit of the tested word with equal probability. Our main contribution is a combinatorial property defined on the Tanner graph associated with the code tester ("well-structured tester"). We show that a family of codes is smoothly locally testable if and only if it has a well-structured tester.As a case study we show that the standard tester for the Hadamard code is "well-structured," giving an alternative proof of the local testability of the Hadamard code, originally proved by (Blum, Luby, Rubinfeld, J. Comput. Syst. Sci. 47 (1993) 549-595) (STOC 1990). Additional connections to the works of (Ben-Sasson, Harsha, Raskhodnikova, SIAM J. Comput 35 (2005) 1-21) (SICOMP 2005) and of (Lachish, Newman and Shapira, Comput Complex 17 (2008) 70-93) are also discussed.

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

Kopparty

Meir

Ron-Zewi

et al. 2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity exp(Õ( √ log n)). Previously such codes were known to exist only with Ω(n β ) query complexity (for constant β > 0), and there were several, quite different, constructions known.Our codes are based on a general distance-amplification method of Alon and Luby [AL96]. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of sub-constant relative distance.Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(Õ( √ log n)), which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with exp(Õ( √ log n)) query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any o(n) query complexity.Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters. * A preliminary version of this work appeared as [Mei14].

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Strong LTCs with Inverse Polylogarithmic Rate and Soundness

Cited by 4 publications

References 17 publications

Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

Strong LTCs with Inverse Poly-Log Rate and Constant Soundness

A combinatorial characterization of smooth LTCs and applications

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

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