  This paper characterizes the trade-offs between information and energy
transmission over an additive white Gaussian noise channel in the finite
block-length regime with finite sets of channel input symbols. These trade-offs
are characterized using impossibility and achievability bounds on the
information transmission rate, energy transmission rate, decoding error
probability (DEP) and energy outage probability (EOP) for a finite block-length
code. Given a set of channel input symbols, the impossibility results identify
the tuples of information rate, energy rate, DEP and EOP that cannot be
achieved by any code using the given set of channel inputs. A novel method for
constructing a family of codes that satisfy a target information rate, energy
rate, DEP and EOP is also proposed. The achievability bounds identify the set
of tuples of information rate, energy rate, DEP and EOP that can be
simultaneously achieved by the constructed family of codes. The proposed
construction matches the impossibility bounds for the information rate, energy
rate, and the EOP. However, for a given information rate, energy rate and EOP,
the achieved DEP is higher than the impossibility bound due to the choice of
the decoding sets made during the code construction.
