Abelian Group Codes for Classical and Classical-Quantum Channels: One-shot and Asymptotic Rate Bounds

TL;DR

This paper investigates the limits of classical and classical-quantum group codes for communication channels, providing new bounds on their performance in both one-shot and asymptotic regimes.

Contribution

It introduces a novel input distribution for group codes, characterizes their performance using hypothesis testing, and establishes tight bounds on group capacities.

Findings

Derived a new one-shot performance characterization for group codes.

Established single-letter lower bounds on asymptotic group capacities.

Provided matching upper bounds for asymptotic group capacities.

Abstract

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.