On the rectangle method in proofs of robustness of tensor products

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. So far, a few important families of tensor products were shown to be robust, and a counter-example of a product that is not robust was also given. However, a precise characterization of codes whose tensor product is robust is yet unknown.In this work, we highlight a common theme in the previous works on the subject, which we call “the rectangle method”. In short, we observe that all proofs of robustness in the previous works are done by constructing a certain “rectangle”, while in the counter-example no such rectangle can be constructed. We then show that a rectangle can be constructed if and only if the tensor product is robust, and therefore the proof strategy of constructing a rectangle is complete.

► We consider the problem of whether the tensor product of two codes is robust.
► We observe that all previous works on the subject have share a common strategy.
► We prove that this strategy is complete, and can be used without loss of generality.