Quantum LDPC Codes With Almost Linear Minimum Distance

We give a construction of quantum LDPC codes of dimension (N) and distance (N/ N) as the code length N. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance (N^1- /2/ N) and dimension (N^ N), where 0. 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 there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least R with, in some sense, optimal circulant size (N/ N) as the code length N.