Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

James Chin-Jen Pang, Sandeep Pradhan, Hessam Mahdavifar

TL;DR

The paper develops a unified one-shot and asymptotic framework for Abelian group codes over classical and classical-quantum channels. By introducing an input distribution aligned with the encoding homomorphism and employing hypothesis-testing relative entropy, it derives tight one-shot achievability and converse bounds for both classical and CQ settings. In the asymptotic regime, it yields single-letter expressions and matching upper bounds for CQ channels, demonstrating when group codes achieve, or fail to achieve, channel capacity. Through detailed examples, the work highlights the interplay between group structure, coset decomposition, and achievable rates, illustrating both the power and limitations of group codes in varied channel models.

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.

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

TL;DR

The paper develops a unified one-shot and asymptotic framework for Abelian group codes over classical and classical-quantum channels. By introducing an input distribution aligned with the encoding homomorphism and employing hypothesis-testing relative entropy, it derives tight one-shot achievability and converse bounds for both classical and CQ settings. In the asymptotic regime, it yields single-letter expressions and matching upper bounds for CQ channels, demonstrating when group codes achieve, or fail to achieve, channel capacity. Through detailed examples, the work highlights the interplay between group structure, coset decomposition, and achievable rates, illustrating both the power and limitations of group codes in varied channel models.

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.
Paper Structure (32 sections, 11 theorems, 170 equations, 3 figures)

This paper contains 32 sections, 11 theorems, 170 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical and CQ Channel Model
  4. Groups and Group Codes
  5. Definition of Achievability for Classical Channel Coding
  6. Definition of Achievability for CQ Channel
  7. Hypothesis testing relative entropy
  8. Abelian Group Code Ensemble
  9. Abelian Groups
  10. The Image Ensemble
  11. The $H_{\hat{\theta}}$ coset
  12. Regular versus irregular codes
  13. One-shot Classical Group Coding
  14. Achievability
  15. Generalized Achievability
  16. ...and 17 more sections

Key Result

Lemma 1

For a group $\tilde{G}$, let $\phi:J\rightarrow \tilde{G}$ be a homomorphism from some group $J$ to $\tilde{G}$. Then $\phi(J)\leqslant \tilde{G}$. Moreover, for any subgroup $\tilde{H}$ of $\tilde{G}$ there exists a corresponding group $J$ and a homomorphism $\phi:J\rightarrow \tilde{G}$ such that

Figures (3)

  • Figure 1: Numerical evaluation of the one-shot quantum information rate $I_{\textnormal{H}}^{\epsilon}(X;Y)$ for certain range of values of $\epsilon$.
  • Figure 2: Numerical evaluation of the one-shot quantum information rate $I_{\textnormal{H}}^{\epsilon} (\overline{X};[X]Y)$ for certain range of values of $\epsilon$: $\hat{\theta}_{22}=0$ and $\hat{\theta}_{22}=1$.
  • Figure 3: Codes over the decimal numbers for a symmetric channel: Group codes achieve the capacity while linear codes over $\mathbb{Z}_7$ do not.

Theorems & Definitions (35)

  • Example 1
  • Lemma 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Example 2
  • Example 3
  • Definition 4
  • Theorem 1
  • ...and 25 more