On the rectangle method in proofs of robustness of tensor products

Or Meir
2012 Information Processing Letters  
Given two error correcting codes R, C, their tensor product R ⊗ C is the error correcting code that consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The code R ⊗ C is said to be robust if, for every matrix M that is far from R⊗C, it holds that the rows and columns of M are far from R and C respectively. Ben-Sasson and Sudan (ECCC TR04-046) asked under which conditions the product R ⊗ C is robust. So far, a few important families of tensor products
more » ... shown to be robust, and a counterexample 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 counterexample 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.
doi:10.1016/j.ipl.2011.11.007 fatcat:vjpz4r2khreh7fbal4ldfziw4q