We investigate the coboundary expansion property of tensor product codes, known as product expansion, which plays an important role in recent constructions of good quantum LDPC codes and classical locally testable codes. Prior research has shown that this property is equivalent to agreement testability and robust testability for products of two codes with linear distance. However, for products of more than two codes, product expansion is a strictly stronger property. In this paper, we prove that a collection of an arbitrary number of random codes over a sufficiently large field has good product expansion. We believe that, in the case of four codes, the same ideas can be used to construct good quantum locally testable codes, in a way similar to the current constructions that use only products of two codes.

翻译：我们研究了张量积码的上边缘扩展性质，即乘积扩展性，该性质在近期构建良好量子LDPC码与经典局部可测试码中具有重要作用。已有研究表明，对于两个具有线性距离的码的乘积，该性质等价于一致性可测试性与鲁棒可测试性。然而，对于超过两个码的乘积，乘积扩展性是严格更强的性质。本文证明：在充分大域上，任意数量随机码的集合具有良好乘积扩展性。我们相信，在四个码的情形下，类似思路可用于构建良好的量子局部可测试码，其方式类似于当前仅使用两个码乘积的构造。