We present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group $\mathbb{F}_2^n$. Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree $\exp(n^\epsilon)$ for every $\epsilon >0$, improving on a construction by Golowich [Gol23] which achieves $\epsilon =1/2$. [Gol23] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmannian posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDXs of any polynomial degree $\exp(\epsilon n$) for any constant $\epsilon > 0$, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [LMSY23]. Establishing coboundary expansion through Gromov's "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.

翻译：我们提出了两种基于阿贝尔群 $\mathbb{F}_2^n$ 的凯莱高维扩展器（HDX）的新显式构造。我们的扩展性证明仅使用线性代数与组合论证。第一种构造给出了任意常数维度且具有次多项式度 $\exp(n^\epsilon)$（对于每个 $\epsilon >0$）的局部谱 HDX，改进了 Golowich [Gol23] 提出的 $\epsilon =1/2$ 的构造。[Gol23] 通过稀疏化子空间的完全格拉斯曼偏序集来推导这些 HDX。我们构造的新颖之处在于能够稀疏化任何扩展的格拉斯曼偏序集，从而实现迭代稀疏化和更小的度。稀疏格拉斯曼集（在高维扩展器理论中具有独立研究价值）作为凯莱图的生成集。我们的第二种构造给出了任意多项式度 $\exp(\epsilon n)$（对于任意常数 $\epsilon > 0$）的二维 HDX，它同时是谱扩展器和上边缘扩展器。据我们所知，这是首个此类非平凡构造。我们将其命名为约翰逊复形，因为它源于经典的约翰逊方案，其顶点作为该凯莱图的生成集。此构造可视为 [LMSY23] 近期随机几何复形的去随机化版本。通过 Gromov 的“锥方法”及相关等周不等式来建立上边缘扩展性，是该构造中最复杂的部分。尽管这两种构造差异显著，但我们证明它们共享一种共同结构，类似于哈达玛码中向量的交集模式。我们提出了此类“类哈达玛”构造的一般框架，以期能产生新的 HDX。