Maximally Extendable Product Codes are Good Coboundary Expanders

TL;DR

This paper proves that collections of multiple random codes over large fields exhibit strong product expansion properties, which are crucial for constructing advanced quantum and classical error-correcting codes.

Contribution

It demonstrates that an arbitrary number of random codes over large fields have good product expansion, extending previous results limited to two-code products.

Findings

Random codes over large fields have good product expansion.

Product expansion is strictly stronger than agreement and robust testability for multiple codes.

Potential application in constructing quantum locally testable codes with more than two codes.

Abstract

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.