具有常数扩张的立方体层复形及其在渐近好qLTC中的应用

中文摘要 AI 辅助

对于每个固定的整数$r \ge 4$和$2 \le k \le r-2$，我们构造了$r$维立方体层复形，其度-$k$ CSS码具有正的常数速率、线性距离和常数健全性，且行重和列重有界。取$r=4$和$k=2$可得到一族渐近好的二元qLTC。\n 我们构造的核心是范数一求值集上显式Reed-Solomon码的均匀乘积扩张定理。关键在于，当局部码长度增长时，扩张常数保持有界且远离零。我们将这些码置于算术立方体复形上，从而为所得的层及其对偶获得常数局部扩张。结合Dinur、Lin和Vidick（FOCS 2024）的局部到全局框架以及层对偶性，这给出了线性距离和常数健全性，而局部码维度的非对称选择则给出了正速率。所得码是显式的且可在多项式时间内计算。

英文摘要

For every fixed integers $r \ge 4$ and $2 \le k \le r-2$, we construct $r$-dimensional cubical sheaf complexes whose degree-$k$ CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking $r=4$ and $k=2$ gives a family of asymptotically good binary qLTCs. At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.

发表机构

• University of Michigan, Ann Arbor（密歇根大学）
• Carnegie Mellon University（卡内基梅隆大学）
• Tsinghua University（清华大学）
• University of Washington（华盛顿大学）

机构由 AI 辅助整理，请以论文原文为准。