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首页> 外文期刊>IEEE Transactions on Information Theory >The Log-Volume of Optimal Codes for Memoryless Channels, Asymptotically Within a Few Nats
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The Log-Volume of Optimal Codes for Memoryless Channels, Asymptotically Within a Few Nats

机译：无记忆通道的最优代码的对数，渐近地在渐近线内

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摘要

Shannon's analysis of the fundamental capacity limits for memoryless communication channels has been refined over time. In this paper, the maximum volume M*avg(n, ∈) of length-n codes subject to an average decoding error probability ∈ is shown to satisfy the following tight asymptotic lower and upper bounds as n → ∞: A∈ + o(1) ≤ log M*avg(n, ∈) - [nC - √nV∈ Q-1(∈)+ (1/2)log n] ≤ A∈ + o(1), where C is the Shannon capacity, V∈ is the ∈-channel dispersion, or secondorder coding rate, Q is the tail probability of the normal distribution, and the constants AE and AE are explicitly identified. This expression holds under mild regularity assumptions on the channel, including nonsingularity. The gap A∈ - A∈ is one nat for weakly symmetric channels in the Cover-Thomas sense, and typically a few nats for other symmetric channels, for the binary symmetric channel, and for the Z channel. The derivation is based on strong large-deviations analysis and refined central limit asymptotics. A random coding scheme that achieves the lower bound is presented. The codewords are drawn from a capacityachieving input distribution modified by an O(1/√n) correction term.
机译：随着时间的推移，Shannon对无记忆通信通道的基本容量限制的分析已得到完善。在本文中，长度为n的代码的最大容量M * avg（n，∈）受平均解码错误概率∈的约束，表示满足以下紧渐近的上下界，即n→∞：A∈+ o（ 1）≤log M * avg（n，∈）-[nC-√nV∈Q-1（∈）+（1/2）log n]≤A∈+ o（1），其中C是香农容量， V∈是ε信道色散或二阶编码率，Q是正态分布的尾部概率，并且明确标识了常数AE和AE。该表达式在通道的轻微规律性假设（包括非奇异性）下成立。对于Cover-Thomas意义上的弱对称通道，间隙A∈-A∈是一个nat，对于其他对称通道，二进制对称通道和Z通道，间隙通常为几个nat。该推导基于强大的大偏差分析和精确的中心极限渐近性。提出了一种实现下限的随机编码方案。码字是从通过O（1 /√n）校正项修改的容量实现输入分布中得出的。

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