A collection of linear codes has good 'product expansion' when, in their tensor product, any locally inconsistent assignment (small on every axis) is close to a genuine tensor codeword — equivalently, the associated chain complex is a good coboundary expander. This paper proves that an arbitrary number of independent random linear codes over a sufficiently large finite field has good product expansion with high probability, lifting earlier results restricted to two or a few codes. The payoff is a plentiful, flexible supply of ingredients for maximally extendable product codes, which underlie constructions of good quantum LDPC codes and classical locally testable codes from high-dimensional expanders.
This paper studies the coboundary expansion of tensor (product) codes and its role in constructing good quantum LDPC codes and classical locally testable codes. Viewing a linear code as a chain compl…