BEC上多维空间耦合码中有界区域引发的阈值饱和
Threshold Saturation from a Bounded Region in Multidimensional Spatially Coupled Codes over the BEC
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中文总结 AI 辅助
本文证明在BEC上，多维空间耦合LDPC和MN码中，用有界超立方体替换缩短平板可将缩短比例从V^{-1/d}降至V^{-1}，实现阈值饱和，MN码达到容量。
中文摘要 AI 辅助
我们证明了在二进制擦除信道（BEC）上，一个有界的缩短区域能够引发多维空间耦合正则LDPC码和MacKay--Neal（MN）码的整个解码过程。在每一个固定的有限维度中，均匀超立方体耦合允许耦合宽度和缩短区域的选择独立于空间位置的总数$V$。在固定宽度和维度$d>1$时，将一个缩短的平板替换为一个有界超立方体，可以将缩短比例从$V^{-1/d}$量级降低到$V^{-1}$量级，并且在校验计数率界中的缩短项上也有同样的改进。正则LDPC码在其未耦合势阈值以下解码。MN码对于每一个整数度选择$\ell>r\geq2$，$g\geq2$都能达到容量：它们的实际传输速率趋向于$r/\ell$，并且在和积解码下，其平均比特擦除概率在$1-r/\ell$以下消失。该证明结合了端点势恒等式、辅助约束的移除、在方向上一致的有穷时间估计，以及将平坦边界进展转移到扩张球的曲率比较。对于MN码，初等不等式为所有这些度数建立了不动点正性。二维密度演化示例说明了初始缩短区域的影响；一般充分常数未进行数值评估。
英文摘要
We prove that a bounded shortened region initiates decoding throughout multidimensional spatially coupled regular LDPC and MacKay--Neal (MN) codes over the binary erasure channel (BEC). In every fixed finite dimension, uniform hypercube coupling permits the coupling width and shortened region to be chosen independently of the total number $V$ of spatial positions. At fixed widths and dimension $d>1$, replacing a shortened slab by a bounded hypercube reduces the shortening fraction from order $V^{-1/d}$ to order $V^{-1}$, with the same improvement in the shortening term of the check-count rate bound. Regular LDPC codes decode below their uncoupled potential threshold. MN codes achieve capacity for every integer degree choice $\ell>r\geq2$, $g\geq2$: their actual transmitted rates tend to $r/\ell$ and their average bit-erasure probabilities under sum-product decoding vanish below $1-r/\ell$. The proof combines an endpoint-potential identity, removal of an auxiliary constraint, finite-time estimates uniform in direction, and a curvature comparison that transfers flat-boundary progress to expanding balls. For MN codes, elementary inequalities establish fixed-point positivity for all these degrees. Two-dimensional density-evolution examples illustrate the dependence on the initial shortened region; the general sufficient constants are not evaluated numerically.
发表机构
- Institute of Science Tokyo（东京科学大学）
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