  As an application of generalised statistical mechanics, it is studied a
possible route toward a consistent generalised information theory in terms of a
family of non-extensive, non-parametric entropies $H^\pm_D(P)$. Unlike other
proposals based on non-extensive entropies with a parameter dependence, our
scheme is asymptotically equivalent to the one formulated by Shannon, while it
differs in regions where the density of states is reasonably small, which leads
to information distributions constrained to their background. Two basic
concepts are discussed to this aim. First, we prove two effective coding
theorems for the entropies $H^\pm_D(P)$. Then we calculate the channel capacity
of a binary symmetric channel (BSC) and a binary erasure channel (BEC) in terms
of these entropies. We found that processes such as data compression and
channel capacity maximisation can be improved in regions where there is a low
density of states, whereas for high densities our results coincide with
Shannon's formulation.
