MacKay-Neal码在MAP译码下达到信道容量
MacKay-Neal Codes Achieve Capacity under MAP Decoding
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中文总结 AI 辅助
本文证明MacKay-Neal码在固定度数下，对容量大于$3/\ell$的对称信道，MAP译码的块错误概率为$O(\log N/N)$，达到信道容量。
中文摘要 AI 辅助
统计力学分析预测，MacKay-Neal码在固定度数下可以达到信道容量。我们证明了这一预测对于每个固定整数$\ell\ge4$的非耦合$(\ell,3,3)$ MN socket系综成立。对于每个容量严格大于$3/\ell$的二元输入无记忆对称信道，在最大后验（MAP）译码下，系综平均块错误概率为$O(\log N/N)$，其中$N$是传输块长度。实际传输速率收敛到$3/\ell$。该结果同时包含打孔和传输变量，并且不要求稀疏方阵可逆。证明通过解析论证建立了六个比特在偶校验约束下的熵不等式，该论证利用对称性缩减定义域，将其内部驻点（两个偏导数均为零的点）限制在对角线上，并通过显式多项式界控制所得标量函数。精确的配置计数随后在二元对称信道上界定了条件熵。通过互信息的标准比较将熵界扩展到一般对称信道；独立的输出擦除和传输码的最小距离估计产生比特和块错误界。
英文摘要
Statistical-mechanical analyses predict that MacKay-Neal codes can achieve channel capacity at fixed degrees. We prove this prediction for the uncoupled $(\ell,3,3)$ MN socket ensemble for every fixed integer $\ell\ge4$. For each binary-input memoryless symmetric channel of capacity strictly greater than $3/\ell$, the ensemble-average block error probability under maximum a posteriori (MAP) decoding is $O(\log N/N)$, where $N$ is the transmitted blocklength. The actual transmitted rate converges to $3/\ell$. The result includes both punctured and transmitted variables and does not condition the sparse square matrix on invertibility. The proof establishes an entropy inequality for six bits subject to even parity by an analytic argument that reduces the domain by symmetry, restricts its interior stationary points, where both partial derivatives vanish, to the diagonal, and controls the resulting scalar functions by explicit polynomial bounds. Exact configuration counts then bound the conditional entropy on the binary symmetric channel. A standard comparison by mutual information extends the entropy bound to general symmetric channels; independent output erasures and a minimum-distance estimate for the transmitted code yield the bit and block error bounds.
