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Locally Testable Codes and PCPs of AlmostLinear Length
, 2002
"... Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using ..."
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Cited by 69 (22 self)
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Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using
Some Applications of Coding Theory in Computational Complexity
, 2004
"... Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory ..."
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Cited by 50 (2 self)
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Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory and to cryptography.
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 7 (5 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
Sound 3query PCPPs are long
, 2008
"... We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot ..."
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Cited by 6 (2 self)
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We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the prooflength and show that any verifier obtaining “best possible” soundness must query an exponentially long proof. In terms of techniques, we focus on the special class of inspective verifiers that read at most 2 proofbits per invocation. For such verifiers we prove exponential lengthsoundness tradeoffs that are later on used to imply our main results for the case of general (i.e., not necessarily inspective) verifiers. To prove the exponential tradeoff for inspective verifiers we show a connection between PCPP proof length and propertytesting query complexity, that may be of independent interest. The connection is that any linear property that can be verified with proofs of length ℓ by linear inspective verifiers must be testable with query complexity ≈ log ℓ.
Short Locally Testable Codes and Proofs (Survey)
, 2005
"... We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs. Locally testability refers to approximately testing large objects based on a very small number of probes, each retrieving a single bit in the represe ..."
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Cited by 2 (0 self)
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We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs. Locally testability refers to approximately testing large objects based on a very small number of probes, each retrieving a single bit in the representation of the object. This yields superfast approximatetesting of the corresponding property (i.e., be a codeword or a valid proof). We also review the related concept of local decodable codes. The currently best results regarding locally testable codes and proofs demonstrate a tradeoff between the number of probes and the length of the code or proof (relative to the information that it encodes). Actually, the length is always &quot;nearly linear&quot;, and the tradeoff is between number of probes and the &quot;level of nearlinearity&quot;. Needless to say, it is not clear whether this tradeoff is inherent. The survey consists of two independent (i.e., selfcontained) parts that cover the same material at different levels of rigor and detail. Still, in spite of the repetitions, there may be a benefit in reading both parts.
Locally Testable Codes and PCPs of AlmostLinear Length
, 2004
"... Abstract We initiate a systematic study of locally testable codes; that is, errorcorrecting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject noncodewords with probab ..."
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Abstract We initiate a systematic study of locally testable codes; that is, errorcorrecting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject noncodewords with probability proportional to their distance from the code. Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques that we develop for the construction of the latter to yield new PCPs. Our main results are locally testable codes and PCPs of almostlinear length. Specifically, we present: ffl Locally testable binary (linear) codes in which k information bits are encoded by a codeword of length approximately k \Delta exp(plog k). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet. ffl PCP systems of almostlinear length for SAT. The length of the proof is approximately n \Delta exp(plog n) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length n1+O(1=q) for verification by q queries. Keywords: ErrorCorrecting Codes, PCP, randomized reductions, lowdegree tests, codeword tests,
Locally Testable Codes and PCPs of AlmostLinear Length
, 2006
"... Abstract We initiate a systematic study of locally testable codes; that is, errorcorrecting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject noncodewords with probab ..."
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Abstract We initiate a systematic study of locally testable codes; that is, errorcorrecting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject noncodewords with probability proportional to their distance from the code. Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques that we develop for the construction of the latter to yield new PCPs. Our main results are locally testable codes and PCPs of almostlinear length. Specifically, we prove the existence of the following constructs: ffl Locally testable binary (linear) codes in which k information bits are encoded by a codeword of length k \Delta exp ( ~O(plog k)). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet. ffl PCP systems of almostlinear length for SAT. The length of the proof is n\Delta exp ( ~O(plog n)) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length n1+O(1=q) for verification by q queries. The novel techniques in use include a random projection of certain codewords and PCPoracles that preserves localtestability, an adaptation of PCP constructions to obtain &quot;linear PCPoracles &quot; for proving conjunctions of linear conditions, and design of PCPs with some new soundness properties. Keywords: ErrorCorrecting Codes, PCP, randomized reductions, lowdegree tests, codeword tests, the Probabilistic Method