Locally testable codes via high-dimensional expanders
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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.
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