We expose a strong connection between good $2$-query locally testable codes (LTCs) and high dimensional expanders. Here, an LTC is called good if it has constant rate and linear distance. Our emphasis in this work is on LTCs testable with only $2$ queries, which are of particular interest to theoretical computer science. This is done by introducing a new object called a sheaf that is put on top of a high dimensional expander. Sheaves are vastly studied in topology. Here, we introduce sheaves on simplicial complexes. Moreover, we define a notion of an expanding sheaf that has not been studied before. We present a framework to get good infinite families of $2$-query LTCs from expanding sheaves on high dimensional expanders, utilizing towers of coverings of these high dimensional expanders. Starting with a high dimensional expander and an expanding sheaf, our framework produces an infinite family of codes admitting a $2$-query tester. We show that if the initial sheaved high dimensional expander satisfies some conditions, which can be checked in constant time, then these codes form a family of good $2$-query LTCs. We give candidates for sheaved high dimensional expanders which can be fed into our framework, in the form of an iterative process which conjecturally produces such candidates given a high dimensional expander and a special auxiliary sheaf. (We could not verify the prerequisites of our framework for these candidates directly because of computational limitations.) We analyse this process experimentally and heuristically, and identify some properties of the fundamental group of the high dimensional expander at hand which are sufficient (but not necessary) to get the desired sheaf, and consequently an infinite family of good $2$-query LTCs.

翻译：我们揭示了良好$2$查询局部可测试码与高维扩展子之间的强关联。此处，称局部可测试码为良好当且仅当其具有常数率与线性距离。本文重点研究仅需$2$次查询即可测试的局部可测试码，这类码对理论计算机科学具有特殊意义。为此，我们引入一种称为"层"的新结构，置于高维扩展子之上。拓扑学中已有对层的广泛研究。本文则引入单纯复形上的层概念，并首次定义了扩展层这一新概念。我们提出一个框架，利用高维扩展子的覆盖塔结构，从这些扩展子上的扩展层生成良好$2$查询局部可测试码的无限族。以高维扩展子和扩展层为起点，该框架可生成一个允许$2$次查询测试的无穷码族。我们证明：若初始赋层高维扩展子满足某些可在常数时间内验证的条件，则这些码构成良好$2$查询局部可测试码族。我们给出可输入该框架的赋层高维扩展子候选，通过一个迭代过程实现——该过程在给定高维扩展子和特殊辅助层时，可推测性地生成此类候选。（由于计算限制，我们无法直接验证这些候选是否满足框架前提条件。）我们通过实验与启发式分析该过程，并识别了相关高维扩展子基本群的若干充分（但非必要）性质，这些性质可确保所需层的存在，进而生成无穷族良好$2$查询局部可测试码。