High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

翻译：高维扩充器同时满足谱和组合（上同调）扩展性质，近期在PCP和编码理论中取得了重大突破，但此类复形的唯一已知构造极其复杂，需要深入的代数数论知识。在本工作中，我们基于标志复形（子空间链）的投影，给出了一种极其简单的次多项式度数的组合构造，该构造满足：(i) 局部谱扩充器，(ii) 上同调扩充器，以及(iii) 交换上同调扩充器。作为推论，我们还首次给出了在“1%”区间内具有良好一致性测试的近线性大小的组合超图，以及一个具有近线性大小的简单PCP构造。