Computability of the Channel Reliability Function and Related Bounds1

The channel reliability function is an important tool that characterizes the reliable transmission of messages over communication channels. For many channels, only upper and lower bounds of the function are known. We analyze the computability of the reliability function and its related functions. We show that the reliability function is not a Turing computable performance function. The same also applies to the functions of the sphere packing bound and the expurgation bound. Furthermore, we show that the R
∞
function is not Banach Mazur computable and additive.