On Good $2$-Query Locally Testable Codes from Sheaves on High Dimensional Expanders

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.