Locally Testing Direct Product in the Low Error Range

Dinur, Irit; Goldenberg, Elazar

doi:10.1109/focs.2008.26

Cited by 24 publications

(22 citation statements)

References 8 publications

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“…We give a 3-query test which, if passed by C with probability q > exp(−k 1/3 ), certifies that C can be approximately list-decoded, and in particular one codeword in the list approximately agrees with a q-fraction of the entries in C. Our techniques (see below) also allow us to considerably simplify the analysis of the 2-query test of [DG08] (for poly(k)-agreement).…”

Section: Our Resultsmentioning

confidence: 99%

“…We solve it here for their 2-query test in the list-decoding regime! We show that, for any k, there is a polynomial family of k-tuples, such that if C passes the 2-query test of [DG08] with probability q > 1/k α then it must have poly(q)-agreement with an approximate direct-product codeword. In coding language, we provide a locally testable, list-decodable k-fold direct-product code of polynomial rate.…”

Section: Our Resultsmentioning

confidence: 99%

“…The possibility of a derandomized 2-query test (even in the unique decoding regime) was raised in [DG08] as an open question. We solve it here for their 2-query test in the list-decoding regime!…”

Section: Our Resultsmentioning

confidence: 99%

“…The first result to test the direct-product code in the list-decoding regime was obtained a few months ago by Dinur and Goldenberg [DG08] (building on the earlier work by Feige and Kilian [FK00]). They give a 2-query test which, if C passes with probability q > 1/k α (for some fixed α > 0), certifies that C can be approximately list-decoded as above.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

“…This current work draws much from these, and adapts them to the testing problem. As explained above, in the testing world one wants to certify what is given as an assumption in the decoding world, and so this adaptation is sometimes impossible (as the [DG08] counterexample shows) and sometimes possible but intricate. But many of the technical notions and lemmas nevertheless apply here.…”

Section: Our Techniques and Direct-product Decodingmentioning

confidence: 99%

See 4 more Smart Citations

New direct-product testers and 2-query PCPs

Impagliazzo

Kabanets

Wigderson

2009

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

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The "direct product code" of a function f gives its values on all k-tuples (f (x 1 ), . . . , f (x k )). This basic construct underlies "hardness amplification" in cryptography, circuit complexity and PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [DG08] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a 2-query test which works for polynomially small success probability 1/k α , and show that no such test works below success probability 1/k.Our main result is a 3-query test which works for exponentially small success probability exp(−k α ). Our techniques (based on recent simplified decoding algorithms for the same code [IJKW08]) also allow us to considerably simplify the analysis of the 2-query test of [DG08]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of k, and success probability 1/k α . Finally we show the applicability of the new tests to PCPs. Starting with a 2-query PCP over an alphabet Σ and with soundness error 1 − δ, Rao [Rao08] (building on Raz's (k-fold) parallel repetition theorem [Raz98] and Holenstein's proof [Hol07]) obtains a new 2-query PCP over the alphabet Σ k with soundness error exp(−δ 2 k). Our techniques yield a 2-query PCP with soundness error exp(−δ √ k); this is incomparable to Rao's result [Rao08] since our bound also applies when k < 1/δ. Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [FK00], but with simpler analysis and exponentially better soundness error. *

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Motivation and Backgroundmentioning

confidence: 99%

Section: Our Techniques and Direct-product Decodingmentioning

confidence: 99%

See 3 more Smart Citations

New direct-product testers and 2-query PCPs

Impagliazzo

Kabanets

Wigderson

2009

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

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The Structure of Winning Strategies in Parallel Repetition Games

Dinur

Goldenberg

2010

Lecture Notes in Computer Science

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Some Recent Results on Local Testing of Sparse Linear Codes

Kopparty

Saraf

2010

Property Testing

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We study the local testability of linear codes. We first reformulate this question in the language of tolerant linearity testing under a non-uniform distribution. We then study the question of linearity testing under non-uniform distributions directly, and give a sufficient criterion for linearity to be tolerantly testable under a given distribution. We show that several natural classes of distributions satisfy this criterion (such as product distributions and low Fourier-bias distributions), thus showing that linearity is tolerantly testable under these distributions. This in turn implies that the corresponding codes are locally testable. For the case of random sparse linear codes, we show the testability and decodability of such codes in the presence of very high noise rates. More precisely, we show that any linear code in F n 2 which is: • sparse (i.e., has only poly(n) codewords) • unbiased (i.e., each nonzero codeword has Hamming weight in (1/2 − n −γ , 1/2 + n −γ) for some constant γ > 0) can be locally tested and locally list decoded from (1/2 −)-fraction errors using only poly(1) queries to the received word. This simultaneously simplifies and strengthens a result of Kaufman and Sudan, who gave a local tester and local (unique) decoder for such codes from some constant fraction of errors. For the case of Dual BCH codes, these algorithms can also be made to run in sublinear time. We also give sub-exponential time algorithms for list-decoding arbitrary unbiased (but not necessarily sparse) linear codes in the high-error regime. * The current extended abstract is a summary of some of the results that appeared in "Tolerant linearity testing and locally testable codes" that appeared in RANDOM 2009 and "Local list-decoding and testing of random linear codes from high-error" that will appear in STOC 2010.

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Locally Testing Direct Product in the Low Error Range

Cited by 24 publications

References 8 publications

New direct-product testers and 2-query PCPs

New direct-product testers and 2-query PCPs

The Structure of Winning Strategies in Parallel Repetition Games

Some Recent Results on Local Testing of Sparse Linear Codes

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