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.

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.

We give a construction of quantum LDPC codes of dimension$\Theta (\log N)$and distance$\Theta (N/\log N)$as the code length$N\to \infty $. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance$\Omega (N^{1-\alpha /2}/\log N)$and dimension$\Omega (N^\alpha \log N)$, where$0 \le \alpha < 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed$R < 1$there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least$R$with, in some sense, optimal circulant size$\Omega (N/\log N)$as the code length$N\to \infty $.

Quasi-cyclic (QC) codes are a wide class of error-correcting codes possessing nice theoretical properties and having many practical applications. This paper provides a new approach to the problem of efficient encoding of QC codes based on the Chinese remainder theorem (CRT). We present a fast systematic CRT-based encoding algorithm that has superior asymptotic complexity than the previous methods based on shift registers. We also consider the encoding problem for QC low-density parity-check (LDPC) codes. In the special case when the parity part of a sparse parity-check QC matrix has a QC generalized inverse we propose a systematic CRT-based encoding algorithm that can exploit the parity-check matrix sparseness. We also give necessary and sufficient conditions when a QC matrix over an arbitrary field has a QC generalized inverse of the same circulant size.