The exact order of the optimal sub-exponentially decaying factor in the classical bounds on the error probability of fixed-length codes over a Gallager-symmetric discrete memoryless channel with and without ideal feedback is determined for rates above the critical rate. Regardless of the availability of feedback, it is shown that the order of the optimal sub-exponential factor exhibits a dichotomy. Moreover, the proof technique is used to establish the third-order term in the normal approximation for symmetric channels, where a similar dichotomy is shown to exist.

中文翻译：

---

关于信道编码中错误概率的精确渐近：对称信道

对于高于临界速率的速率，确定在有和没有理想反馈的 Gallager 对称离散无记忆信道上固定长度代码的错误概率的经典界限中最佳次指数衰减因子的确切顺序。无论反馈的可用性如何，都表明最佳次指数因子的顺序表现出二分法。此外，证明技术用于在对称通道的正态近似中建立三阶项，其中显示存在类似的二分法。