In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. In contrast, our construction is a Cayley complex over the group $\mathbb{F}_2^k$, with Cayley generating set given by a Grassmannian HDX. Our construction is in part motivated by a coding-theoretic interpretation of Grassmannian HDXs that we present, which provides a formal connection between Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this interpretation to prove a general characterization of the 1-homology groups over $\mathbb{F}_2$ of Cayley simplicial complexes over $\mathbb{F}_2^k$. Using this result, we construct simplicial complexes on $N$ vertices with arbitrarily good local expansion for which the dimension of the 1-homology group grows as $\Omega(\log^2N)$. No prior constructions in the literature have been shown to achieve as large a 1-homology group.

翻译：本文提出一种新的单纯复形构造方法，该构造具有次多项式度数且具备任意良好的局部谱扩张性质。此前，唯一已知具有任意良好扩张性且度数低于多项式的高维扩展图（HDXs）基于两种构造之一：拉马努金复形和陪集复形。相比之下，我们的构造是基于群$\mathbb{F}_2^k$的凯莱复形，其凯莱生成集由格拉斯曼HDX给出。本文的构造部分受我们提出的格拉斯曼HDX编码理论解释所启发，这种解释建立了格拉斯曼HDX、单纯HDX与低密度奇偶校验（LDPC）码之间的形式化联系。我们应用该解释证明了$\mathbb{F}_2$上关于$\mathbb{F}_2^k$的凯莱单纯复形的一阶同调群的通用刻画定理。利用这一结果，我们在$N$个顶点上构造了具有任意良好局部扩张性的单纯复形，其一阶同调群维数达到$\Omega(\log^2N)$。现有文献中的构造尚未能实现如此高维数的一阶同调群。