The Tensor Product of Two Codes Is Not Necessarily Robustly Testable

There has been significant interest lately in the task of constructing codes that are testable with a small number of random probes. Ben-Sasson and Sudan show that the repeated tensor product of codes leads to a general class of locally testable codes. One question that is not settled by their work is the local testability of a code generated by a single application of the tensor product. Special cases of this question have been studied in the literature in the form of “tests for bivariate polynomials”, where the tensor product has been shown to be locally testable for certain families of codes. However the question remained open for the tensor product of generic families of codes. Here we resolve the question negatively, giving families of codes whose tensor product does not have good local testability properties.