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16
Robust Locally Testable Codes and Products of Codes
 In Proc. RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science
, 2004
"... We continue the investigation of locally testable codes, i.e., errorcorrecting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed ..."
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We continue the investigation of locally testable codes, i.e., errorcorrecting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing lowdegree multivariate polynomials (which are a special case of tensor codes).
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 ..."
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Cited by 15 (3 self)
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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.
Locally Testable Codes Require Redundant Testers
, 2010
"... Locally testable codes (LTCs) are errorcorrecting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give errorcorrecting codes whose duals have (superlinearly) many ..."
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Cited by 8 (4 self)
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Locally testable codes (LTCs) are errorcorrecting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give errorcorrecting codes whose duals have (superlinearly) many small weight codewords. Examining this feature appears to be one of the promising approaches to proving limitation results for (i.e., upper bounds on the rate of) LTCs. Unfortunately till now it was not even known if LTCs need to be nontrivially redundant, i.e., need to have one linear dependency among the lowweight codewords in its dual. In this paper we give the first lower bound of this form, by showing that every positive rate constant query strong LTC must have linearly many redundant lowweight codewords in its dual. We actually prove the stronger claim that the actual test itself must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, nonredundant (in fact, low redundancy) local testing is impossible. Our main theorem is a special case of a more general theorem that applies to any tester for an arbitrary linear locally testable code C. The general theorem can be used, for instance, to provide an arguably
The tensor product of two good codes is not necessarily locally testable, ECCC
, 2007
"... 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 res ..."
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Cited by 6 (3 self)
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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. Addressing this question, Paul Valiant (APPROXRANDOM 2005) constructed two linear codes with constant relative distance whose tensor product is not robust. However, one of those codes has a subconstant rate. We show that this construction can be modified such that both codes have both constant rate and constant relative distance. We also provide an alternative proof for the nonrobustness of the tensor product of those codes, based on the inverse direction of the “rectangle method ” that was presented by the second author (ECCC TR07061). We believe that this proof gives an additional intuition for why this construction works. 1
LIST DECODING TENSOR PRODUCTS AND INTERLEAVED CODES
"... Abstract. We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. • We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rat ..."
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Abstract. We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. • We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. • We show that for every code, its list decoding radius remains unchanged under mwise interleaving for an integer m. This generalizes a recent result of Dinur et al. [6], who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). • Using the notion of generalized Hamming weights, we give better list size bounds for both tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of bounding the list size to bounding the number of closeby lowrank codewords. For decoding linear transformations, using rankreduction together with other ideas, we obtain list size bounds that are tight over small fields. Our results give better bounds on the list decoding radius than what is obtained from the Johnson bound, and yield rather general families of codes decodable beyond the Johnson bound. 1.
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 ..."
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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.
Testing Linear Properties: Some general themes
, 2011
"... The last two decades have seen enormous progress in the development of sublineartime algorithms — i.e., algorithms that examine/reveal properties of “data” in less time than it would take to read all of the data. A large, and important, subclass of such properties turn out to be “linear”. In partic ..."
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The last two decades have seen enormous progress in the development of sublineartime algorithms — i.e., algorithms that examine/reveal properties of “data” in less time than it would take to read all of the data. A large, and important, subclass of such properties turn out to be “linear”. In particular, these developments have contributed to the rich theory of probabilistically checkable proofs (PCPs) and locally testable codes (LTCs). In this survey, we focus on some of the general technical themes at work behind the many results in this area.