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Nested Symmetric Polar Codes

Marvin Rübenacke, Andreas Zunker, Felix Krieg, Stephan ten Brink

TL;DR

Th theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED, and a flexible polar code design for automorphism ensemble successive cancellation decoding is constructed.

Abstract

In this paper, we propose a data-driven algorithm to design rate- and length-flexible polar codes. While the algorithm is very general, a particularly appealing use case is the design of codes for automorphism ensemble decoding (AED), a promising decoding algorithm for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC) applications. To this end, theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED. Using the proposed algorithms, such a flexible polar code design for automorphism ensemble successive cancellation (SC) decoding is constructed, outperforming existing code designs for AED and also the 5G polar code under cyclic redundancy check (CRC)-aided successive cancellation list (SCL) decoding.

Nested Symmetric Polar Codes

TL;DR

Th theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED, and a flexible polar code design for automorphism ensemble successive cancellation decoding is constructed.

Abstract

In this paper, we propose a data-driven algorithm to design rate- and length-flexible polar codes. While the algorithm is very general, a particularly appealing use case is the design of codes for automorphism ensemble decoding (AED), a promising decoding algorithm for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC) applications. To this end, theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED. Using the proposed algorithms, such a flexible polar code design for automorphism ensemble successive cancellation (SC) decoding is constructed, outperforming existing code designs for AED and also the 5G polar code under cyclic redundancy check (CRC)-aided successive cancellation list (SCL) decoding.

Paper Structure

This paper contains 20 sections, 4 theorems, 10 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Polar Codes
  5. Partial Order
  6. Automorphisms of Polar Codes
  7. Successive Cancellation Decoding
  8. Automorphism Ensemble Decoding
  9. Nested Polar Codes
  10. Finding Rate Compatible Sequences
  11. General Methodology
  12. Rate Compatible Symmetric Polar Codes
  13. Nested Sequences
  14. Sequential Construction
  15. Disambiguation
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\mathcal{P}(N,K)$ be a polar code with block profile $\boldsymbol{s} = [s_1,\dots,s_m ]$. Then its high and low nested subcodes $\mathcal{P}^\ell(N/2,K^\ell)$ and $\mathcal{P}^h(N/2,K^h)$ have at least block profile $\boldsymbol{s}' = [s_1,\dots,s_m-1 ]$ each (with $\boldsymbol{s}' = [s_1,\dots

Figures (2)

  • Figure 1: Graph of polar code designs with performance metric $\mu$. The optimal sequence $\mathcal{S}$ minimizes the area under the graph.
  • Figure 2: Required SNR for BLER $\le 10^{-3}$ vs. code dimension $K$ for different block lengths.

Theorems & Definitions (15)

  • Definition 1: Automorphism Group
  • Definition 2: Affine Permutation
  • Definition 3: Stabilizer
  • Definition 4: Low and High Nested Subcodes
  • Proposition 1: Symmetric Subcodes
  • Proposition 2: Symmetric Supercodes
  • Example 1
  • Definition 5: Optimal Sequence
  • Proposition 3: Optimality
  • Definition 6: Zero-Padded Polar Code
  • ...and 5 more