Results 1 
6 of
6
Combinatorial Construction of Locally Testable Codes
, 2009
"... An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. While the best known construction of LTCs by BenSasson and Sudan (STOC 2005) and Dinu ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. While the best known construction of LTCs by BenSasson and Sudan (STOC 2005) and Dinur (J. ACM 54(3)) achieves very e cient parameters, it relies heavily on algebraic tools and on PCP machinery. In this work we present a new and arguably simpler construction of LTCs that is purely combinatorial, does not rely on PCP machinery and matches the parameters of the best known construction. However, unlike the latter construction, our construction is not entirely explicit.
www.theoryofcomputing.org Tensor Products of Weakly Smooth Codes are Robust ∗
, 2008
"... Abstract: We continue the study of robustly testable tensor codes and expand the class of base codes that can be used as a starting point for the construction of locally testable codes via robustly testable tensor products. In particular, we show that all uniqueneighbor expander codes and all local ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract: We continue the study of robustly testable tensor codes and expand the class of base codes that can be used as a starting point for the construction of locally testable codes via robustly testable tensor products. In particular, we show that all uniqueneighbor expander codes and all locally correctable codes, when tensored with any other gooddistance code, are robustly testable and hence can be used to construct locally testable codes. Previous work by Dinur et al. (2006) required stronger expansion properties to obtain locally testable codes. Our proofs follow by defining the notion of weakly smooth codes that generalize the smooth codes of Dinur et al. We show that weakly smooth codes are sufficient for constructing robustly testable tensor codes. Using the weaker definition, we are able to expand the family of base codes to include the aforementioned ones. ACM Classification: E.4 AMS Classification: 68Q99 Key words and phrases: Linear code, tensor code, expander code
On the Rectangle Method in proofs of Robustness of Tensor Products
, 2011
"... Given two codes R and C, their tensor product R ⊗ C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R ⊗ C is said to be robust if for every matrix M that is far from R ⊗ C it holds that the rows and columns of M are far on average from R and C ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Given two codes R and C, their tensor product R ⊗ C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R ⊗ C is said to be robust if for every matrix M that is far from R ⊗ C it holds that the rows and columns of M are far on average from R and C respectively. BenSasson and Sudan (ECCC TR04046) have asked under which conditions the product R ⊗ C is robust. So far, a few important families of tensor products were shown to be robust, and a counterexample of a product that is not robust was also given. However, a precise characterization of codes whose tensor product is robust is yet unknown. In this work, we highlight a common theme in the previous works on the subject, which we call “the rectangle method”. In short, we observe that all proofs of robustness in the previous works are done by constructing a certain “rectangle”, while in the counterexample no such rectangle can be constructed. We then show that a rectangle can be constructed if and only if the tensor product is robust, and therefore the proof strategy of constructing a rectangle is complete.
On the Rectangle Method in proofs of Robustness of Tensor Products∗ Or Meir†
, 2011
"... Given two codes R and C, their tensor product R⊗C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R⊗C is said to be robust if for every matrix M that is far from R⊗C it holds that the rows and columns of M are far on average from R and C respe ..."
Abstract
 Add to MetaCart
(Show Context)
Given two codes R and C, their tensor product R⊗C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R⊗C is said to be robust if for every matrix M that is far from R⊗C it holds that the rows and columns of M are far on average from R and C respectively. BenSasson and Sudan (ECCC TR04046) have asked under which conditions the product R ⊗ C is robust. So far, a few important families of tensor products were shown to be robust, and a counterexample of a product that is not robust was also given. However, a precise characterization of codes whose tensor product is robust is yet unknown. In this work, we highlight a common theme in the previous works on the subject, which we call “the rectangle method”. In short, we observe that all proofs of robustness in the previous works are done by constructing a certain “rectangle”, while in the counterexample no such rectangle can be constructed. We then show that a rectangle can be constructed if and only if the tensor product is robust, and therefore the proof strategy of constructing a rectangle is complete. 1
Highrate locallycorrectable and locallytestable codes with subpolynomial query complexity∗
, 2015
"... In this work, we construct the first locallycorrectable codes (LCCs), and locallytestable codes (LTCs) with constant rate, constant relative distance, and subpolynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can eve ..."
Abstract
 Add to MetaCart
In this work, we construct the first locallycorrectable codes (LCCs), and locallytestable codes (LTCs) with constant rate, constant relative distance, and subpolynomial 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(nβ) query complexity (for constant β> 0), and there were several, quite different, constructions known. Our codes are based on a general distanceamplification 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 nonstandard regime of subconstant relative distance. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(Õ(