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

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.