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.

翻译：本文刻画了在有限块长和有限信道输入符号集下，加性高斯白噪声信道中信息与能量传输之间的权衡关系。这些权衡通过有限块长码的信息传输速率、能量传输速率、译码错误概率（DEP）和能量中断概率（EOP）的不可达界与可达界来表征。给定一组信道输入符号，不可达结果识别了无法通过任何使用该给定信道输入符号集的码达到的信息速率、能量速率、DEP和EOP的元组。论文还提出了一种构建满足目标信息速率、能量速率、DEP和EOP的码族的新方法。可达界确定了所构建码族能够同时实现的信息速率、能量速率、DEP和EOP的元组集合。所提出的构造方法在信息速率、能量速率和EOP方面与不可达界相匹配。然而，对于给定的信息速率、能量速率和EOP，由于码构造过程中译码集合的选择，实际达到的DEP高于不可达界。