Reed-Muller码的最优检验
Optimal Testing of Reed-Muller Codes
摘要：
我们考虑一个给定的函数f：F_2^n -> F_2是否接近n元任意d次多项式的检验问题，也称为Reed-Muller检验问题。Gowers范数基于此属性的自然2^{d+1}-查询测试。Alon等人[AKKLR 05]重新发现了这个测试，并表明它接受概率为1的所有d次多项式，而拒绝概率为Omega（1）-far的函数。我们给出了这个检验的渐近最优分析，并表明它拒绝（甚至只有）Omega（2^{-d}）-far具有Omega（1）-概率的函数（因此拒绝概率是独立于d和n的普适常数）。这意味着函数的（d+1）st Gowers范数与其与d次多项式的最大相关性之间的紧密关系，当相关性接近1时。我们的证明是通过对n的归纳来进行的，并且甚至对经典的Blum-Luby-Rubinfeld [BLR 93]线性检验进行了新的分析，用于设置将F_2^n映射到F_2的函数。最优性来自对反例的更严格分析，以[GT 09，LMS 08]构建的“Gowers范数的逆猜想”。我们的结果有几个含义。首先，它表明Gowers范数检验是宽容的，因为它也接受封闭码字。其次，它改进了Viola和Wigderson [VW 07]给出的多项式异或引理的参数。第三，它意味着一个“查询层次”的结果，仿射不变属性的属性测试。也就是说，对于每个函数q（n），它给出了一个仿射不变的性质，可以用O（q（n））-查询来测试，但不能用O（q（n））-查询来测试，补充了[GKNR 09]关于图性质的类似结果。
英文摘要：
We consider the problem of testing if a given function f : F_2^n -> F_2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural 2^{d+1}-query test for this property. Alon et al. [AKKLR05] rediscovered this test and showed that it accepts every degree d polynomial with probability 1, while it rejects functions that are Omega(1)-far with probability Omega(1/(d 2^{d})). We give an asymptotically optimal analysis of this test, and show that it rejects functions that are (even only) Omega(2^{-d})-far with Omega(1)-probability (so the rejection probability is a universal constant independent of d and n). This implies a tight relationship between the (d+1)st Gowers norm of a function and its maximal correlation with degree d polynomials, when the correlation is close to 1. Our proof works by induction on n and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping F_2^n to F_2. The optimality follows from a tighter analysis of counterexamples to the "inverse conjecture for the Gowers norm" constructed by [GT09,LMS08]. Our result has several implications. First, it shows that the Gowers norm test is tolerant, in that it also accepts close codewords. Second, it improves the parameters of an XOR lemma for polynomials given by Viola and Wigderson [VW07]. Third, it implies a "query hierarchy" result for property testing of affine-invariant properties. That is, for every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [GKNR09] for graph properties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
