发表机构
Institute of Science Tokyo（东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明空间耦合MacKay-Neal码在固定度数下于BMS信道达到容量，通过势能符号分析给出度数二和三的精确结果。
AI 中文摘要
我们在二进制输入无记忆对称信道上建立了空间耦合MacKay-Neal码的两个与度数相关的结果。这些码本采用均匀随机平滑，并在活动链外部缩短两类变量节点。对于$r=g=3$，每个整数$\ell\geq4$，以及每个容量大于$R=r/\ell$的信道，存在一个码实现序列，其实际传输速率趋于$R$，且平均和积比特错误趋于零。对于$r=g=2$和每个$\ell\in\{3,4,5\}$，存在一个二进制对称信道区间，其容量大于$R$，但在终止密度演化下，随着链长相对于耦合宽度增长，传输比特错误保持为正。这两个结果均源于同一密度值势能在非耦合不动点处的符号。度数三的证明结合了$\ell\geq33$的解析界和$4\leq\ell\leq32$的精确区间证书，随后通过阈值饱和及实际速率的投影论证完成。度数二的证明解析地构造了一个具有负势能的不动点，并控制了边界贡献。我们将该势能和度数二的分岔条件与早期的统计力学预测联系起来。有限证书和验证软件可在版本化补充材料中获取。
英文摘要
We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For $r=g=3$, every integer $\ell\geq4$, and each channel of capacity greater than $R=r/\ell$, there is a sequence of code realizations whose actual transmitted rate tends to $R$ and whose average sum-product bit error tends to zero. For $r=g=2$ and each $\ell\in\{3,4,5\}$, an interval of binary symmetric channels has capacity greater than $R$ but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for $\ell\geq33$ with exact interval certificates for $4\leq\ell\leq32$, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.
Comments43 pages; expanded preliminaries, proof explanations, BEC comparison, and historical context; exact certificates and verification software: https://github.com/kasaikenta/mn-bms-capacity-certificates/releases/tag/v1.1.1
