Given two codes $R$ and $C$, their tensor product $R \otimes C$ consists of all matrices whose rows are codewords of $R$ and whose columns are codewords of $C$. The product $R \otimes C$ is said to be robust if for every matrix $M$ that is far from $R \otimes C$ it holds that the rows and columns of $M$ are far on average from $R$ and $C$ respectively. Ben-Sasson and Sudan (ECCC TR04-046) have asked under which conditions the product $R \otimes C$ 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 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.

Given two error correcting codes $R$,$C$, their tensor product $R\otimes C$ 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 $R\otimes C$ is said to be robust if, for every matrix $M$ that is far from $R \otimes C$, 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 $R \otimes C$ 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 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.

Given two error correcting codes $R$,$C$, their tensor product $R\otimes C$ 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 $R\otimes C$ is said to be robust if, for every matrix $M$ that is far from $R \otimes C$, 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 $R \otimes C$ 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 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.

Given linear two codes R,C, their tensor product R\otimes C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R\otimes C is said to be robust if for every matrix M that is far from R\otimes C it holds that the rows and columns of M are far from R and C respectively. Ben-Sasson and Sudan (ECCC TR04-046) have asked under which conditions the product R\otimes C is robust. During the last few years, 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 remains unknown.

In this note we highlight a common theme in the above papers, which we call ``The Rectangle Method''. In short, we observe that all proofs of robustness in the above papers are done by constructing a ``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.

Given linear two codes R,C, their tensor product $R \otimes C$

consists of all matrices whose rows are codewords of R and whose

columns are codewords of C. The product $R \otimes C$ is said to

be robust if for every matrix M that is far from $R \otimes C$

it holds that the rows and columns of M are far from R and C

respectively. Ben-Sasson and Sudan (ECCC TR04-046) have asked under

which conditions the product $R \otimes C$ is robust. During the last

few years, 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 remains unknown.

In this note we highlight a common theme in the above papers, which

we call "The Rectangle Method". In short, we observe that all

proofs of robustness in the above papers are done by constructing

a "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.