Locally Testing Direct Products in the High Error Range
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چکیده
Given a function f : X → Σ, its `-wise direct product is the function F = f ` : X` → Σ` defined by F (x1, . . . , x`) = (f(x1), . . . , f(x`)). In this paper we study the local testability of the direct product encoding (mapping f 7→ f `). Namely, given an arbitrary function F : X` → Σ`, we wish to determine how close it is to f ` for some f : X → Σ, by making the smallest possible number of random queries into F (namely, two). This question has first been studied by Goldreich and Safra and later the following simple two-query test has been studied by Dinur and Reingold: Choose a random pair x,x′ ∈ X` that have m coordinates in common. Accept iff F (x) and F (x′) agree on the common coordinates. Dinur and Reingold showed that if the test accepts with sufficiently high probability (close to 1) then F is close to f ` for some f . In this work we analyze the case of low acceptance probability of the test. We show that even if the test passes with small probability, ε > 0, already F must have a non-trivial structure and in particular must agree with some f ` on nearly ε of the domain. Moreover, we give a structural characterization of all functions F on which the test passes with probability ε. We find a list of functions f1, . . . , ft such that essentially the only way T ′ will accept on a pair x,x′, is if both F (x) and F (x′) agree with fi. This is related to approximate local-decoding of this code, as studied by Impagliazzo et. al. Our result means that both the testing and the approximate local decoding can be done in “one shot” with the minimal possible number (only two) of queries. Our results hold for values of ε as small as `−Ω(1), and we show that below 1/` no characterization is possible. ∗Weizmann Institute. Research supported by ISF and BSF grants. [email protected] †Hebrew University. [email protected]
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تاریخ انتشار 2008