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# l17 - 6.896 Sublinear Time Algorithms Lecture 17 Lecturer...

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6.896 Sublinear Time Algorithms September 11, 2004 Lecture 17 Lecturer: Eli Ben-Sasson Scribe: Rafael Pass 1 Lower Bounds for Property Testing: Rest of the Proof This lecture is the continuation of the previous one, in which we prove the following Theorem, stated last time: Theorem 1 ([1]) There exists integers c, d , and values δ > 0 , 0 < < 1 / 2 , γ > 0 such that for all suﬃciently large n , with high probability, for a random low density parity check code (LDPC) every ( δ, 0 , , q ) linear test requires q γn queries. As mentioned in the previous lecture, this result also extends to any test, and not just linear tests. Towards the goal of proving the above theorem we rely on Yao’s minmax principle, which (in our case) boils down to showing one “bad” distribution over inputs, such that every deterministic linear test fails with probability larger than . We continue to use the notation and definitions from previous lecture. 1.1 The “bad” distribution Recall that the code V defined by the constraint matrix A is V = ker( A ) = { x F n 2 | Ax = 0 } . Let B i be the uniform distribution over strings that falsify constraint A i , but satify the rest, i.e., B i = ker( A i ) ker( A ), where A i is the matrix obtained by removing row i from A . Define the “bad” distribution to be the uniform convex combination of B i ’s. In other words, the “bad” distribution is obtained by uniformly chosing i [ m ] (recall m is the number of rows in A ) and then uniformly chosing an element in B i . Note that it follows from the definition of B that for every w B , w / V . In fact, every w B is “far” from V , as is shown in the following lemma (a proof sketch of the lemma is given in the next section).

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l17 - 6.896 Sublinear Time Algorithms Lecture 17 Lecturer...

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