Quantum LDPC Codes with Almost Linear Minimum Distance
| Autor: | Panteleev, Pavel, Kalachev, Gleb |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1109/TIT.2021.3119384 |
| Popis: | 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 $N\to\infty$. Comment: 17 pages, 2 figures. Accepted for publication in IEEE Transactions on Information Theory |
| Databáze: | arXiv |
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