Maximally Extendable Product Codes are Good Coboundary Expanders
High-dimensional quantum LDPC codes and classical locally testable codes (LTCs) suffer from insufficient expansion, limiting their local testability and robustness. Method: We systematically investigate upper-edge expansion in multi-code (≥4) product constructions, integrating combinatorial coding theory, cohomological analysis, and random code ensemble techniques. Contribution/Results: We establish, for the first time, that large random code families exhibit strong product expansion; moreover, we prove that multi-code product expansion strictly surpasses bipartite (two-code) expansion—a novel structural property. We further derive an exact quantitative trade-off between expansion and local testability, thereby providing both theoretical guarantees and a constructive pathway toward the first explicit four-code quantum LTC. This framework significantly enhances local testability and robustness of quantum LDPC codes.