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On Polar Coding with Feedback

Ling Liu, Qi Cao, Liping Li, Baoming Bai

TL;DR

The paper addresses finite-length performance limitations of polar codes by introducing polar feedback coding with genie-aided SC (GA-SC) decoding, leveraging a relaxed construction threshold $\epsilon_{\mathrm{th}}^* = 1/\log N$ to boost rate without sacrificing reliability. It develops a probabilistic framework that models the number of error-inducing indices $|\mathcal{T}|$ via a negative-binomial process, enabling precise predictions of block error rate and decoding delay through parameters estimated from $\mathsf{E}[|\mathcal{T}|]$ and $\mathsf{Var}[|\mathcal{T}|]$. The analysis yields practical insights and is supported by simulations across BSC, BEC, and BIAWGN channels, showing notable finite-length gains and accurate performance forecasting. The work also outlines extensions, including error-index compression and GA-$T_{\max}$ decoding, highlighting a pathway to integrate feedback into polar-code design for improved finite-length performance in real systems.

Abstract

In this work, we investigate the performance of polar codes with the assistance of feedback in communication systems. Although it is well known that feedback does not improve the capacity of memoryless channels, we show that the finite length performance of polar codes can be significantly improved as feedback enables genie-aided decoding and allows more flexible thresholds for the polar coding construction. To analyze the performance under the new construction, we then propose an accurate characterization of the distribution of the error event under the genie-aided successive cancellation (SC) decoding. This characterization can be also used to predict the performance of the standard SC decoding of polar codes with rates close to capacity.

On Polar Coding with Feedback

TL;DR

The paper addresses finite-length performance limitations of polar codes by introducing polar feedback coding with genie-aided SC (GA-SC) decoding, leveraging a relaxed construction threshold to boost rate without sacrificing reliability. It develops a probabilistic framework that models the number of error-inducing indices via a negative-binomial process, enabling precise predictions of block error rate and decoding delay through parameters estimated from and . The analysis yields practical insights and is supported by simulations across BSC, BEC, and BIAWGN channels, showing notable finite-length gains and accurate performance forecasting. The work also outlines extensions, including error-index compression and GA- decoding, highlighting a pathway to integrate feedback into polar-code design for improved finite-length performance in real systems.

Abstract

In this work, we investigate the performance of polar codes with the assistance of feedback in communication systems. Although it is well known that feedback does not improve the capacity of memoryless channels, we show that the finite length performance of polar codes can be significantly improved as feedback enables genie-aided decoding and allows more flexible thresholds for the polar coding construction. To analyze the performance under the new construction, we then propose an accurate characterization of the distribution of the error event under the genie-aided successive cancellation (SC) decoding. This characterization can be also used to predict the performance of the standard SC decoding of polar codes with rates close to capacity.
Paper Structure (9 sections, 3 theorems, 20 equations, 7 figures, 2 tables)

This paper contains 9 sections, 3 theorems, 20 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Codes and the SK Coding Scheme
  3. Polar Codes for Channel Coding
  4. The SK Feedback Coding
  5. Polar Feedback Coding
  6. The Coding Scheme
  7. Performance Analysis
  8. Simulation Results
  9. Conclusion

Key Result

Theorem 1

For the proposed ideal feedback model, the optimal construction of polar codes in terms of coding rate is given by: and $\mathcal{I}=\mathcal{F}^c$, where $\frac{1}{\log N}$ is called the optimal threshold $\epsilon^*_{\mathop{\mathrm{\mathrm{th}}}\nolimits}$.

Figures (7)

  • Figure 1: A geometric interpretation of the SK coding scheme.
  • Figure 2: Block diagram of the polar feedback coding scheme.
  • Figure 3: A statistic perspective of the occurrence of error events.
  • Figure 4: The fitness of the negative binomial distribution for $|\mathcal{T}|$.
  • Figure 5: $\frac{\mathsf{Var}[|\mathcal{T}|]}{\mathsf{E}[|\mathcal{T}|]}$ for different BMSCs and $N$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3