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.
