The tensor product of two good codes is not necessarily robustly testable

Given two codes R and C  , their tensor product R⊗CR⊗C consists of all matrices whose rows are codewords of R and whose columns are codewords of C  . The product R⊗CR⊗C is said to be robust if for every matrix M   that is far from R⊗CR⊗C it holds that the rows and columns of M are far on average from R and C   respectively. Ben-Sasson and Sudan (RSA 28(4), 2006) have asked under which conditions the product R⊗CR⊗C is robust.Addressing this question, Paul Valiant (APPROX-RANDOM 2005) constructed two linear codes with constant relative distance whose tensor product is not robust. However, one of those codes has a sub-constant rate. We show that this construction can be modified such that both codes have both constant rate and constant relative distance. We also provide an alternative proof for the non-robustness of the tensor product of those codes, based on the inverse direction of the “rectangle method” that was presented by the second author (ECCC TR07-061). We believe that this proof gives an additional intuition for why this construction works.

► We consider the problem of whether the tensor product of two good codes is robust.
► We generalize an example of Paul Valiant to give a negative example.
► We give a new proof to the correctness of Valiantʼs example.