发表机构
Bilkent University（比尔肯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对变长反馈码，提出与可达界相同系数的二阶逆界，确立二阶基本极限，并刻画最优码的必要性质及二元擦除信道的最小期望解码时间。
AI 中文摘要
我们研究在平均解码时间和错误概率约束下，离散无记忆信道上的变长反馈（VLF）码。在非消失错误概率机制中，Polyanskiy、Poor 和 Verdú（2011）推导了最大可达码本大小的对数的可达性和逆界。这些界确立了 ε-容量，但在二阶展开中留下了阶为 log N 的间隙，其中 N 是平均解码时间。Yavas 和 Tan（2025）将可达性界中 log N 的系数从 -1 改进为 -C/C1，其中 C 是信道容量，C1 是两个条件输出分布之间的最大 Kullback-Leibler 散度。我们推导了具有相同系数的逆界，为每个具有有限 C1 的正容量离散无记忆信道确立了二阶基本极限。该结果还涵盖了中等偏差和误差指数机制，包括多项式衰减的错误概率。该逆界使用 Rényi 熵和外在 Jensen-Shannon 散度。我们还推导了渐近最优 VLF 码的必要性质。一阶最优码必须具有提前停止分支，二阶最优码还必须表现出通信和确认行为。最后，对于二元擦除信道，我们确定了每个消息集大小和允许错误概率下的精确最小期望解码时间。
英文摘要
We study variable-length feedback (VLF) codes over a discrete memoryless channel under average decoding-time and error-probability constraints. In the non-vanishing error probability regime, Polyanskiy, Poor, and Verdú (2011) derive achievability and converse bounds on the logarithm of the maximum achievable codebook size. These bounds establish the $ε$-capacity but leave an order-$\log N$ gap in the second-order expansion, where $N$ is the average decoding time. Yavas and Tan (2025) improve the coefficient of $\log N$ in the achievability bound from $-1$ to $-\frac{C}{C_1}$, where $C$ is the channel capacity and $C_1$ is the largest Kullback--Leibler divergence between two conditional output distributions. We derive a converse with the same coefficient, establishing the second-order fundamental limit for every positive-capacity discrete memoryless channel with finite $C_1$. The result also covers the moderate-deviations and error-exponent regimes, including polynomially decaying error probabilities. The converse uses Rényi entropy and the extrinsic Jensen--Shannon divergence. We also derive necessary properties of asymptotically optimal VLF codes. First-order-optimal codes must have an early-stopping branch, and second-order-optimal codes must additionally exhibit communication and confirmation behavior. Finally, for the binary erasure channel, we determine the exact minimum expected decoding time for every message-set size and admissible error probability.
Comments26 pages, submitted to Information Theory Transactions
