Two-sided Robustly Testable Codes

TL;DR

This paper proves that the tensor product of two random linear codes is highly likely to be robustly testable, which has implications for quantum LDPC codes and understanding code properties.

Contribution

It demonstrates that the tensor product of two random linear codes is robustly testable with high probability, advancing the theory of code robustness and applications in quantum error correction.

Findings

Tensor product of two random codes is robustly testable with high probability.

Product of a code with its dual is not robustly testable.

Introduces the concept of product-expansion related to code robustness.

Abstract

We show that the tensor product of two random linear codes is robustly testable with high probability. This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously robustly testable. Such two-sided robustly testable codes (with a much weaker form of robustness) were the key ingredient in the recent constructions of asymptotically good quantum LDPC codes, which ensured their linear minimum distance. We hope that the existence of such codes with a stronger form of robustness, shown here, can be used to simplify the proofs and provide better distance bounds in these constructions. We also give new very simple examples of non-robustly testable codes. We show that if the parity-checks of two codes are mutually orthogonal, then their product is not robustly testable. In particular, this implies that the product of a code with its dual can never be robustly testable. We also study a property of a collection of linear codes called product-expansion, which can be viewed as a coboundary expansion of the cochain complex naturally associated with the product of these codes. We show that this property is related with the robust testability and the agreement testability of the products of codes.