Quantum LDPC Codes With Almost Linear Minimum Distance
Pavel Panteleev, Gleb Vyacheslavovich Kalachev · IEEE Transactions on Information Theory · 2021
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 $.