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Asymmetric Evaluations of Erasure and Undetected Error Probabilities

机译：擦除和未检测到的错误概率的不对称评估

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摘要

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analyzing the ensemble performance in terms of a tradeoff between the code rate, the undetected and total errors. This tradeoff is parameterized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of . In this case, the total error probability is asymptotically a positive constant, while the undetected error probability decays as for some . The proof techniques involve the applications of a modified (or shifted) version of the Gärtner–Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.
机译：带有擦除选项的通道编码问题已针对离散的无记忆通道进行了讨论。表征了码率，未检测到和总错误概率之间的相互作用。使用信息频谱方法，设计了增加块长的代码序列来说明这种折衷。此外，对于具有均匀输入分布的加性离散无记忆通道，我们确定我们的分析相对于整体平均而言是严格的。这是通过在编码率，未检测到的错误和总错误之间进行权衡来分析整体性能来完成的。在一般似然比测试中，此折衷由阈值参数化。研究了两种渐近状态。首先，编码速率趋向于以比对应于中等偏差机制慢的速率增加信道的容量。在这种情况下，两个错误概率都会以次指数和非对称方式衰减。确定了精确的衰减率。其次，编码率倾向于以的速率进行容量传输。在这种情况下，总错误概率渐近地为一个正常数，而未检测到的错误概率就象某些一样衰减。证明技术涉及Gärtner-Ellis定理的修改（或移位）版本和类型类枚举器方法的应用，以表征一系列累积量生成函数的渐近行为。

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