Sparse affine-invariant linear codes are locally testable
نویسندگان
چکیده
منابع مشابه
Limits on the Rate of Locally Testable Affine-Invariant Codes
Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, “algebraic property testing” is not well-understood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was rem...
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Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field Fqn to the subfield Fq and include all properties that form an Fq-vector space and are invariant under affine t...
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We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and “lifts”. The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The “lift” is a less natural property that has been studied before. Previously such results were known for “single-orbit c...
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We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an a ne test. That is, we consider LTCs over a nite eld F, with codeword testers that only use tests of the form avi + bvj = c, where v is the tested word and a, b, c ∈ F. We show that such LTCs, with high minimal distance, must be of constant size. Speci cally, we show that every 2-query LT...
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We initiate a study of “universal locally testable codes” (universal-LTCs). These codes admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. More precisely, a universal-LTC C : {0, 1} → {0, 1} for a family of functions F = { fi : {0, 1} → {0, 1} } i∈[M ] is a code such that for every i ∈ [M ] the subcode {C(x) : fi(x) = 1} is lo...
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ژورنال
عنوان ژورنال: computational complexity
سال: 2015
ISSN: 1016-3328,1420-8954
DOI: 10.1007/s00037-015-0115-6