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Shortened Polar Codes under Automorphism Ensemble Decoding

Charles Pillet, Ilshat Sagitov, Valerio Bioglio, Pascal Giard

TL;DR

The automorphism group of shortened polar codes, designed according to two existing shortening patterns, are shown to be limited but non-empty, making the Automorphism Ensemble (AE) decoding of shortened polar codes possible.

Abstract

In this paper, we propose a low-latency decoding solution of shortened polar codes based on their automorphism groups. The automorphism group of shortened polar codes, designed according to two existing shortening patterns, are shown to be limited but non-empty, making the Automorphism Ensemble (AE) decoding of shortened polar codes possible. Extensive simulation results for shortened polar codes under AE are provided and are compared to the SC-List (SCL) algorithm. The block-error rate of shortened polar codes under AE matches or beats SCL while lowering the decoding latency.

Shortened Polar Codes under Automorphism Ensemble Decoding

TL;DR

The automorphism group of shortened polar codes, designed according to two existing shortening patterns, are shown to be limited but non-empty, making the Automorphism Ensemble (AE) decoding of shortened polar codes possible.

Abstract

In this paper, we propose a low-latency decoding solution of shortened polar codes based on their automorphism groups. The automorphism group of shortened polar codes, designed according to two existing shortening patterns, are shown to be limited but non-empty, making the Automorphism Ensemble (AE) decoding of shortened polar codes possible. Extensive simulation results for shortened polar codes under AE are provided and are compared to the SC-List (SCL) algorithm. The block-error rate of shortened polar codes under AE matches or beats SCL while lowering the decoding latency.
Paper Structure (13 sections, 3 theorems, 7 equations, 4 figures)

This paper contains 13 sections, 3 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. GN-coset codes definition and decoding
  3. $\mathbf{G}_N$-Coset Sub-Families
  4. Affine Automorphism Group of $\mathbf{G}_N$-Coset Codes
  5. Decoders of $\mathbf{G}_N$-Coset Codes
  6. Code Shortening
  7. Group Properties of Shortened Polar Codes
  8. Polar-like Codes and $\mathbf{G}_N$-Coset Mother Codes
  9. Affine Automorphism Group of Mother Codes
  10. Analysis For All Shortening Parameters $(S,K')$
  11. Action of Permutations
  12. Simulation Results
  13. Conclusions

Key Result

Proposition 1

The $\mathbf{G}_N$-coset mother code $\mathcal{C}_{\text{M}}$ of any block or BR shortened polar code is not a polar-like code.

Figures (4)

  • Figure 1: Representation of sub-families of $\mathbf{G}_N$-coset codes.
  • Figure 2: Classification of $\mathcal{C}_{\text{M}}^{\text{BR}}$ and $\mathcal{C}_{\text{M}}^{\text{block}}$ for $N=256$ and a design of $0$ dB. Blue (green) positions correspond to $\mathcal{C}_{\text{M}}^{\text{BR}}\in\mathcal{C}_{\text{large}}$$\left(\mathcal{C}_{\text{M}}^{\text{block}}\in\mathcal{C}_{\text{large}}\right)$, purple positions to $\mathcal{C}_{\text{M}}^{\text{BR}}\wedge\mathcal{C}_{\text{M}}^{\text{block}}\in\mathcal{C}_{\text{large}}$, and white positions to $\mathcal{C}_{\text{M}}^{\text{BR}}\wedge\mathcal{C}_{\text{M}}^{\text{block}}\notin\mathcal{C}_{\text{large}}$.
  • Figure 3: of $(115,51,\mathcal{Z})$ shortened polar codes.
  • Figure 4: Average execution time to decode $(115,51)$ code.

Theorems & Definitions (11)

  • Definition 1: $\mathbf{G}_N$-coset codes
  • Definition 2: Polar codes
  • Definition 3: Reed-Muller codes
  • Definition 4: Polar-like codes
  • Definition 5: Shortened polar codes
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • ...and 1 more