Quantum LDPC Codes With Almost Linear Minimum Distance

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 $
.