@MISC{Dinur08locallytesting, author = {Irit Dinur and Elazar Goldenberg}, title = {Locally Testing Direct Products in the High Error}, year = {2008} }

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Abstract

Given a function f: X → Σ, its ℓ-wise direct product is the function F = f ℓ: X ℓ → Σ ℓ defined by F (x1,..., xℓ) = (f(x1),..., f(xℓ)). In this paper we study the local testability of the direct product encoding (mapping f ↦ → f ℓ). Namely, given an arbitrary function F: X ℓ → Σ ℓ, we wish to determine how close it is to f ℓ 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 ℓ 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 ℓ 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 ℓ 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 ℓ −Ω(1) , and we show that below 1/ℓ no characterization is possible.