CIF: Small: Moderate Deviations and Exact Asymptotics in Information Theory

信息论描述了可靠通信和数据压缩的可能性，它通常通过渐近结果来实现。虽然现有的渐近结果是正确的庆祝，他们有两个局限性。首先，这些结果都不是针对渐近制度，这是最实际的重要。在信道编码的情况下，这是差错概率趋于零并且速率随着块长度的增加而接近容量的状态。 第二，现有的结果往往是太粗糙，是有用的基准，为实际的代码，因为他们隐藏的滋扰因素，可能是显着的，在中等blocklengths.This研究将解决这两个问题。该项目将表征最佳信道码的性能，在该制度中，速率接近容量和错误概率同时趋于零。其次，这将是通过建立更精细的“精确渐近”的界限，在这两个政权和经典的滋扰因素的特点。这些结果也将被扩展到数据压缩和多用户问题，导致更精确地理解编码的基本限制在实际制度的利益。除了推进信息论领域，这项研究将加强其与编码理论和概率论的联系，前者通过为中等块长度提供更有用的性能界限，后者通过两个领域之间的交叉思想。教育部分将修改信息理论课程，以便纳入最近的研究成果，包括该项目的研究成果，并使更多的人能够了解这一主题。

Information theory describes what is possible in reliable communication and data compression, and it generally does so via asymptotic results. While existing asymptotic results are rightly celebrated, they have two limitations. First, none of these results is directed at the asymptotic regime that is the most practically important. In the case of channel coding, this is the regime in which the error probability tends to zero and the rate approaches capacity as the blocklength increases. Second, existing results are often too coarse to be useful as benchmarks for practical codes because they hide nuisance factors that could be significant at moderate blocklengths.This research will address both of these issues. The project will characterize the performance of optimal channel codes for the regime in which the rate approaches capacity and the error probability simultaneously tends to zero. This will be followed by characterizing the nuisance factors in both this regime and the classical ones by establishing more refined "exact-asymptotic" bounds. These results will also be extended to data compression and multiuser problems, leading to a more precise understanding of the fundamental limits of coding in practical regimes of interest.In additional to advancing the field of information theory, this research will strengthen its connections to coding theory and probability theory, the former by providing more useful performance bounds for moderate blocklengths and the latter by cross-fertilizing ideas between the two fields. The educational component will revamp course offerings in information theory in order to allow the inclusion of recent research results, including those from this project, and to make the subject accessible to a wider audience.