Wireless channels are considered that change over time but remain constant for a certain (coherence) period. This behavior is perfectly captured by block fading channels and affects the performance of the corresponding wireless communication systems. Desired closed-form characterizations of optimal transmission schemes remain unknown in many cases. This paper approaches this issue from a fundamental, algorithmic point of view by studying whether or not it is in principle possible to construct or find such optimal transmission schemes algorithmically (without putting any constraints on the computational complexity of such algorithms). To this end, the concept of averaged channels is considered as a model for block fading and it is shown that, although the averaged channel itself is computable, the corresponding capacity need not be computable, i.e., there exists no (universal) algorithm that takes the channel as an input and computes the corresponding capacity expression. Subsequently, examples of block fading channels are presented for which it is even impossible to find an algorithm that computes for every blocklength the corresponding optimal transmission scheme.