Reliability of classical and classical-quantum channels

The performance of a channel is usually measured in terms of its capacity C, intended as the largest rate achievable by block codes with probability of error which vanishes in the block-length. For rates R<C, the probability of error for optimal codes decreases exponentially fast with the block-length, and a more detailed measure of the performance of the channel is the so called reliability function E(R), the first order exponent of this error probability. 

The exact determination of E(R) is in general only possible at high rates, while it remains an open problem at low rates. It includes, as a sub-problem, that of determining the the zero-error capacity (also called Shannon capacity of a graph). We will discuss some bounds on E(R) in the context of classical and classical-quantum setting, analyzing their connection with the Lovász theta function and other known quantities. 

Then, if we have time, I would try to support with some examples my belief that this topic is still an excellent source of interesting problems at the intersection of information theory, combinatorics, coding, etc.