We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

中文翻译：

---

来自高维扩展器的显式 SoS 下界

我们构建了一个显式的 3XOR 实例族，这对于平方和层次结构的 $O(\sqrt{\log n})$ 级别来说是困难的。与涉及随机分量的早期构造相比，我们的系统可以在确定性多项式时间内明确构造。我们的构造基于 Lubotzky、Samuels 和 Vishne 设计的高维膨胀器，称为 LSV 复合物或拉马努金复合物，我们的分析基于这些复合物的两个膨胀概念：共收缩膨胀和局部等周不等式格罗莫夫。我们的结构与 Alev、Jeronimo 和最后一位作者的近期工作形成了有趣的对比~（FOCS 2019）。他们表明，变量对应于高维扩展器中的顶点的 3XOR 实例很容易求解。相比之下，