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.

翻译：我们研究乘积码的共边展开性质（称为乘积展开），该性质在近期所有优秀qLDPC码的构建中均发挥了关键作用。已有研究表明，对于具有线性距离的两码乘积，该性质等价于鲁棒可测试性和一致性可测试性。首先，我们证明对于多个具有线性距离的码的乘积，鲁棒可测试性与一致性可测试性等价。其次，我们给出一个由三个具有线性距离的码构成的乘积示例，该乘积具有鲁棒可测试性，但不具有乘积展开性。