We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include: 1. In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible. 2. We propose a new approach for proving correlation bounds with the central "mod functions", consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results". 3. We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions $(x_{1},\dots,x_{n})\to z^{\sum x_{i}}$ for any $z$ on the complex unit circle; and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction. 4. We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric.

翻译：我们研究了在 $\mathbb{F}_{2}$ 上展示与低次多项式具有小相关性的显式函数这一基本挑战。主要贡献包括：1. 在STOC 2020中，CHHLZ引入了一种证明相关性界的新技术，利用该技术建立了新的低次多项式相关性界，并推测该技术可推广至更高次多项式。我们给出其猜想的反例，实际上排除了更弱的参数条件，证明他们的结果本质上是不可改进的最优结论。2. 我们提出一种证明核心"模函数"相关性界的新方法，包含两个步骤：（I）最大化相关性的多项式是对称的；（II）对称多项式具有小相关性。与文献中相关结果相反，我们推测（I）成立，并论证该方法不受现有"障碍结果"的影响。3. 我们证明了二次多项式的猜想：具体确定了复单位圆上任意 $z$ 对应的二次多项式模2与函数 $(x_{1},\dots,x_{n})\to z^{\sum x_{i}}$ 的最大可能相关性，并证明该最大值由对称多项式实现。为获得结论，我们发展了新的证明技术：将相关性用方向导数表示，并通过逐步限制方向进行分析。4. 我们部分推进了三次多项式的猜想，特别证明了三次部分为对称的三次多项式的紧致相关性界。