An $n$-bit boolean function is resilient to coalitions of size $q$ if any fixed set of $q$ bits is unlikely to influence the function when the other $n-q$ bits are chosen uniformly. We give explicit constructions of depth-$3$ circuits that are resilient to coalitions of size $cn/\log^{2}n$ with bias $n^{-c}$. Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.

翻译：若一个$n$位布尔函数在其余$n-q$位均匀随机选取时，任意$q$个固定位集合对函数值的影响可忽略，则称该函数对规模为$q$的联合攻击具有弹性。我们给出了深度为$3$的电路显式构造，其对规模为$cn/\log^{2}n$的联合攻击具有$n^{-c}$偏置弹性。此前具有相同弹性的显式构造仅具备常数偏置。我们的构造更简洁，并可推广至有偏乘积分布。证明基于前人工作，主要创新在于结合Janson不等式的精细分析，采用了扩展图游走的尾界技术。