We study the problem of transmission of information over classical and classical-quantum channels in the one-shot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity.

翻译：本研究探讨了在单次传输机制下，通过经典及经典-量子信道传输信息的问题，其中所采用的码被约束为群码。在可达性部分，我们提出了一种新的输入概率分布，该分布融合了编码同态映射与底层信道规律。通过随机编码论证，我们以假设检验相对熵量来刻画群码的性能。在逆命题部分，我们基于假设检验方法建立了性能界。进一步地，我们将单次结果应用于渐进平稳无记忆场景，为两类信道建立了群容量的单字母下界。此外，我们还推导了渐进群容量的匹配上界。