Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes

Abstract

We show that cosystolic expansion of sheaves on posets can be derived from local expansion conditions of the sheaf and the poset. When the poset at hand is a cell complex — typically a high dimensional expander — a sheaf may be thought of as generalizing coefficient groups used for defining homology and cohomology, by letting the coefficient group vary along the cell complex. Previous works established local criteria for cosystolic expansion only for simplicial complexes and with respect to constant coefficients. Our main technical contribution is providing a criterion that is more general in two ways: it applies to posets and sheaves, respectively. The importance of working with sheaves on posets (rather than constant coefficients and simplicial complexes) stems from applications to locally testable codes (LTCs). It has been observed by Kaufman–Lubotzky that cosystolic expansion is related to property testing in the context of simplicial complexes and constant coefficients, but unfortunately, this special case does not give rise to interesting LTCs. We observe that this relation also exists in the much more general setting of sheaves on posets. As the language of sheaves is more expressive, it allows us to put this relation to use. Specifically, we apply our criterion for cosystolic expansion in two ways. First, we show the existence of good 2-query LTCs. These codes are actually related to the recent good q-query LTCs of Dinur–Evra–Livne–Lubotzky–Mozes and Panteleev–Kalachev, being the formers’ so-called line codes, but we get them from a new, more illuminating perspective. By realizing these codes as cocycle codes of sheaves on posets, we can derive their good properties directly from our criterion for cosystolic expansion. The local expansion conditions that our criterion requires unfold to the conditions on the “small codes” in Dinur et. al and Panteleev–Kalachev, and hence give a conceptual explanation to why conditions such as agreement testability are required. Second, we show that local testability of a lifted code could be derived solely from local conditions, namely from agreement expansion properties of the local “small” codes which define it. In a work of Dikstein–Dinur–Harsha–Ron-Zewi, it was shown that one can obtain local testability of lifted codes from a mixture of local and global conditions, namely, from local testability of the local codes and global agreement expansion of an auxiliary 3-layer system called a multilayered agreement sampler. Our result achieves the same, but using genuinely local conditions and a simpler 3-layer structure. It is derived neatly from our local criterion for cosystolic expansion, by interpreting the situation in the language of sheaves on posets.