论文标题
量子LDPC代码几乎是线性最小距离
Quantum LDPC Codes with Almost Linear Minimum Distance
论文作者
论文摘要
我们将量子LDPC代码$θ（\ log n）$和距离$θ（n/\ log n）$作为代码长度$ n \ to \ infty $的构造。使用链复合物的产物，该结构还提供了距离$ω（n^{1-α/2}/\ log n）$的量子LDPC代码和尺寸$ω（n^α\ log n）$，其中$ 0 \ leleα<1 $。我们还介绍并研究了一个名为“提升产品”的新操作，该操作自然会概括量子代码和链复合物的产品操作。此外，作为我们在量子代码上的结果的简单副产品，我们在经典代码上获得了新的结果。我们表明，对于任何固定的$ r <1 $，都存在一个渐近良好的经典准环境LDPC LDPC速率代码，至少$ r $，从某种意义上说，最佳的循环尺寸$ω（n/\ log n）$作为代码长度$ n \ to \ suftty $。
We give a construction of quantum LDPC codes of dimension $Θ(\log N)$ and distance $Θ(N/\log N)$ 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 $Ω(N^{1-α/2}/\log N)$ and dimension $Ω(N^α\log N)$, where $0 \le α< 1$. 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 $Ω(N/\log N)$ as the code length $N\to\infty$.
