We study variable-length feedback (VLF) codes over binary-input memoryless symmetric (BMS) channels using posterior matching with small-enough-difference (SED) partitioning. Prior analyses of SED-based schemes rely on bounded log-likelihood ratio (LLR) increments, restricting their scope to discrete-output channels such as the binary symmetric channel (BSC). We remove this restriction and provide an analysis of posterior matching that covers a broad class of BMS channels, including continuous-output channels such as the binary-input AWGN channel. We derive a novel non-asymptotic achievability bound on the expected decoding time that decomposes into communication, confirmation, and recovery terms with explicit dependence on the channel capacity~$C$, the KL divergence~$C_1$, and the Bhattacharyya parameter of the channel. The proof develops new stopping-time and overshoot bounds for submartingales and random walks with unbounded increments, drawing on tools from renewal theory. On the algorithmic side, we propose a low-complexity encoder that enforces the exact SED partition at every step by grouping messages according to their log-likelihood ratios that are assumed to land on a lattice, and applying a batched correction step that restores the partition balance. The resulting encoder complexity is polynomial in the number of transmitted bits. For continuous-output channels, the lattice structure is enforced through output quantization satisfying an exact induced-lattice constraint; the associated capacity loss is $O(\log B / B^2)$ for a $B$-level quantizer. These results yield a VLF coding scheme for BMS channels that simultaneously achieves strong non-asymptotic performance and practical encoder complexity.

翻译：我们研究了在二进制输入无记忆对称（BMS）信道上，采用足够小差异（SED）划分的后验匹配（posterior matching）变长反馈（VLF）码。先前基于SED方案的分析依赖于有界对数似然比（LLR）增量，从而将其适用范围限制在离散输出信道（如二进制对称信道（BSC））上。我们消除了这一限制，并提出了一种后验匹配分析，覆盖了广泛的BMS信道类别，包括连续输出信道（如二进制输入加性白高斯噪声信道）。我们推导了一个新颖的非渐近期望解码时间可达界，该界分解为通信项、确认项和恢复项，并依赖于信道容量$C$、KL散度$C_1$以及信道的巴塔查里亚参数。证明过程发展了针对具有无界增量的子鞅和随机游走的新的停时与超调界限，利用了更新理论中的工具。在算法方面，我们提出了一种低复杂度编码器，该编码器通过假设消息的对数似然比落在格点上对消息进行分组，并应用批量修正步骤恢复划分平衡，从而在每一步强制执行精确的SED划分。所得编码器复杂度与传输比特数成多项式关系。对于连续输出信道，通过满足精确诱导格约束的输出量化来强制实现格点结构；对于$B$级量化器，相关的容量损失为$O(\log B / B^2)$。这些结果为BMS信道提供了一种同时实现强非渐近性能和实际编码器复杂度的VLF编码方案。