# The robustness of the tensor product

**Problem**Given two codes , their

**Tensor Product**is the code that consists of the matrices whose rows are codewords of and whose columns are codewords of . The product is said to be

**robust**if whenever a matrix is far from , the rows (columns) of are far from (, respectively).

The problem is to give a characterization of the pairs whose tensor product is robust.

The question is studied in the context of Locally Testable Codes.

## Bibliography

*[BS] Eli Ben-Sasson, Madhu Sudan, Robust locally testable codes and products of codes, APPROX-RANDOM 2004, pp. 286-297 (See ECCC TR04-046).

[CR] D. Coppersmith and A. Rudra, On the robust testability of tensor products of codes, ECCC TR07-061.

[DSW] Irit Dinur, Madhu Sudan and Avi Wigderson, Robust local testability of tensor products of LDPC codes, APPROX-RANDOM 2006, pp. 304-315 (See ECCC TR06-118).

[GM] Oded Goldreich, Or Meir, The Tensor Product of Two Good Codes Is Not Necessarily Robustly Testable, ECCC TR07-062.

[M] Or Meir, On the Rectangle Method in proofs of Robustness of Tensor Products, ECCC TR07-061.

[V] Paul Valiant, The Tensor Product of Two Codes Is Not Necessarily Robustly Testable, APPROX-RANDOM 2005, pp. 472-481.

* indicates original appearance(s) of problem.

### The formal definition of robustness, and of the problem

In all of the following definitions, the term "distance" refers to "relative Hamming distance".

Given a matrix , let denote the distance from to the nearest codeword of . Let denote the average distance of a row of to , and let denote the average distance of a column of to . Finally, let denote the average of and .

The tensor product is said to be -robust iff for every matrix we have that .

The question is, under what conditions the tensor product is -robust for some constant .

### To be more precise ...

1) When you say codes, do you mean linear codes?

2) What distance you are using when you're saying "far from"?

### To be more precise

1) The question is most interesting for linear codes, but it can also be defined for non-linear codes.

2) The distance is (relative or absolute) Hamming Distance.

## Results in

Eli Ben-Sasson and Michael Viderman. "Composition of semi-LTCs by two-wise Tensor Products" (RANDOM 09)

Eli Ben-Sasson and Michael Viderman. "Tensor Products of Weakly Smooth Codes are Robust" (RANDOM 08)