Composition of semi-LTCs by two-wise tensor products

Abstract

In this paper, we obtain a composition theorem that allows us to construct locally testable codes (LTCs) by repeated two-wise tensor products. This is the first composition theorem showing that repeating the two-wise tensor operation any constant number of times still results in a locally testable code, improving upon previous results which only worked when the tensor product was applied once. To obtain our results, we define a new tester for tensor products. Our tester uses the distribution of the "inner tester" associated with the base code to sample rows and columns of the product code. This construction differs from previously studied testers for tensor product codes which sampled rows and columns uniformly. We show that if the base code is any LTC or any expander code, then the code obtained by taking the repeated two-wise tensor product of the base code with itself is locally testable. In particular, this answers a question posed in the paper of Dinur et al. (2006) by expanding the class of allowed base codes to include all LTCs and not just so-called uniform LTCs whose associated tester queries all codeword entries with equal probability. One corollary of our composition theorem is a simple construction of high-rate LTCs with sublinear query complexity. More formally, we show that for every ?, ? > 0 there exists a family of LTCs over the binary field with query complexity n? and rate at least 1 ? ?.