## Some Improvements to Total Degree Tests (1995)

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Citations: | 42 - 10 self |

### BibTeX

@MISC{Friedl95someimprovements,

author = {Katalin Friedl and Madhu Sudan},

title = { Some Improvements to Total Degree Tests},

year = {1995}

}

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### Abstract

A low-degree test is a collection of simple, local rules for checking the proximity of an arbitrary function to a low-degree polynomial. Each rule depends on the function’s values at a small number of places. If a function satisfies many rules then it is close to a low-degree polynomial. Low-degree tests play an important role in the development of probabilistically checkable proofs. In this paper we present two improvements to the efficiency of low-degree tests. Our first improvement concerns the smallest field size over which a low-degree test can work. We show how to test that a function is a degree d polynomial over prime fields of size only d + 2. Our second improvement shows a better efficiency of the low-degree test of [ 141 than previously known. We show concrete applications of this improvement via the notion of “locally checkable codes”. This improvement translates into better tradeoffs on the size versus probe complexity of probabilistically checkable proofs than previously known.

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Citation Context ...g) > ǫ, then T rejects with probability δ. The low-degree testing problem has been studied widely due to their relationship with probabilistically checkable (holographic) proofs and program checking. =-=[5, 6, 9, 2, 10, 13]-=- study the case of testing the maximum degree and [8, 11, 14, 1] study the case of testing total degree1. Our improvements are to the latter family of testers. We start by describing their testers. De... |

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Citation Context ...lem has been studied widely due to their relationship with probabilistically checkable (holographic) proofs and program checking. [5, 6, 9, 2, 10, 13] study the case of testing the maximum degree and =-=[8, 11, 14, 1]-=- study the case of testing total degree 1 . Our improvements are to the latter family of testers. We start by describing their testers. Definition 2 For points x, h ∈ F m , the line through x with off... |

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Citation Context ...nt to show some sort of robustness. However their proof falls short of showing low-degree tests that work over fields of size d + 2 because of the lack of an “exact characterization” (in the sense of =-=[15]-=-). We complement their work by providing an exact characterization of low-degree polynomials, which shows that their tester is good for prime fields of size d+2, and improves the bound for non-prime f... |

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Citation Context ...lem has been studied widely due to their relationship with probabilistically checkable (holographic) proofs and program checking. [5, 6, 9, 2, 10, 13] study the case of testing the maximum degree and =-=[8, 11, 14, 1]-=- study the case of testing total degree 1 . Our improvements are to the latter family of testers. We start by describing their testers. Definition 2 For points x, h ∈ F m , the line through x with off... |

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Citation Context ...combination of the various proof systems used here. The final ingredient in the proof system is the randomnessefficient parallelization protocol of [1] (which is where the efficiency of the tester of =-=[14]-=- plays a role). Details of the construction will be available in the full paper. Last we would also like to mention two interesting questions that may be raised about locally checkable codes. 1. Does ... |

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