  Locally testable codes (LTC) are error-correcting codes that have a local
tester which can distinguish valid codewords from words that are "far" from all
codewords by probing a given word only at a very few (sublinear, typically
constant) number of locations. Such codes form the combinatorial backbone of
PCPs. A major open problem is whether there exist LTCs with positive rate,
constant relative distance and testable with a constant number of queries.
  In this paper, we present a new approach towards constructing such LTCs using
the machinery of high-dimensional expanders. To this end, we consider the
Tanner representation of a code, which is specified by a graph and a base code.
Informally, our result states that if this graph is part of a high-dimensional
expander then the local testability of the code follows from the local
testability of the base code.
  This work unifies and generalizes the known results on testability of the
Hadamard, Reed-Muller and lifted codes on the Subspace Complex, all of which
are proved via local self correction. However, unlike previous results,
constant rounds of self correction do not suffice as the diameter of the
underlying test graph can be logarithmically large in a high-dimensional
expander and not constant as in all known earlier results. We overcome this
technical hurdle by performing iterative self correction with logarithmically
many rounds and tightly controlling the error in each iteration using
properties of the high-dimensional expander.
  Given this result, the missing ingredient towards constructing a
constant-query LTC with positive rate and constant relative distance is an
instantiation of a base code that interacts well with a constant-degree
high-dimensional expander.
