In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with certain high dimensional expansion relations between its sets. We say that a linear code is modelled over HDE-system, if the collection of linear constraints that the code satisfies could by described via the HDE-system. We show that a code that can be modelled over HDE-system is locally testable. This implies that high dimensional expansion phenomenon solely implies local testability of codes. Prior work had to rely to local notions of local testability to get some global forms of testability (e.g. co-systolic expansion from local one, global agreement from local one), while our work infers global testability directly from high dimensional expansion without relying on some local form of testability. The local testability result that we obtain from HDE-systems is, in fact, stronger than standard one, and we term it amplified local testability. We further show that most of the well studied locally testable codes as Reed-Muller codes and more generally affine invariant codes with single-orbit property fall into our framework. Namely, it is possible to show that they are modelled over an HDE-system, and hence the family of all p-ary affine invariant codes is amplified locally testable. This yields the strongest known testing results for affine invariant codes with single orbit, strengthening the work of Kaufman and Sudan.

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders were derived from the theory of buildings that is far from being elementary. In this work we study high dimensional complexes which are strongly symmetric. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion. It was an open question whether the recent elementary construction of bounded degree spectral high dimensional expanders based on coset complexes give rise to bounded degree cosystolic expanders. In this work we answer this question affirmatively. We show that these complexes give rise to bounded degree cosystolic expanders in dimension two, and that their links are (two-dimensional) coboundary expanders. We do so by exploiting the strong symmetry properties of the links of these complexes using a new machinery developed in this work. Previous works have shown a way to bound the co-boundary expansion using strong symmetry in the special situation of "building like" complexes. Our new machinery shows how to get coboundary expansion for general strongly symmetric coset complexes, which are not necessarily "building like", via studying the (Dehn function of the) presentation of the symmetry group of these complexes.

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.