High-dimensional Expansion of Product Codes is Stronger than Robust and Agreement Testability

TL;DR

This paper investigates the expansion properties of product codes, showing equivalences between robust testability and agreement testability for many codes, and providing an example where robust testability does not imply product expansion.

Contribution

It establishes the equivalence between robust testability and agreement testability for products of many codes and presents a counterexample separating robust testability from product expansion.

Findings

Robust testability for many codes with linear distance is equivalent to agreement testability.

An example of a three-code product that is robustly testable but not product expanding.

Insights into the limitations of product expansion in high-dimensional code constructions.

Abstract

We study the coboundary expansion property of product codes called product expansion, which played a key role in all recent constructions of good qLDPC codes. It was shown before that this property is equivalent to robust testability and agreement testability for products of two codes with linear distance. First, we show that robust testability for product of many codes with linear distance is equivalent to agreement testability. Second, we provide an example of product of three codes with linear distance which is robustly testable but not product expanding.