Composition of Semi-LTCs by Two-Wise Tensor Products

We continue the study of the local testability of error correcting codes constructed by taking the two-wise tensor product of a “base-code” with itself. We show that if the base-code is any locally testable code (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. This extends the results of Dinur et al. in [11] in two ways. First, we answer a question posed in that paper by expanding the class of allowed base-codes to include all locally testable code, and not just so-called

uniform LTCs

whose associated tester queries all codeword entries with equal probability. Second, we show that

repeating

the two-wise tensor operation a 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 the tensor product of LTCs. Our tester uses the distribution of the 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

.