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This paper characterizes the trade-offs between information and energytransmission over an additive white Gaussian noise channel in the finiteblock-length regime with finite sets of channel input symbols. These trade-offsare characterized using impossibility and achievability bounds on theinformation transmission rate, energy transmission rate, decoding errorprobability (DEP) and energy outage probability (EOP) for a finite block-lengthcode. Given a set of channel input symbols, the impossibility results identifythe tuples of information rate, energy rate, DEP and EOP that cannot beachieved by any code using the given set of channel inputs. A novel method forconstructing a family of codes that satisfy a target information rate, energyrate, DEP and EOP is also proposed. The achievability bounds identify the setof tuples of information rate, energy rate, DEP and EOP that can besimultaneously achieved by the constructed family of codes. The proposedconstruction matches the impossibility bounds for the information rate, energyrate, 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 ofthe decoding sets made during the code construction.

Information-Energy Regions in the Finite Block-Length Regime with Finite Channel Inputs