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Precoded Polar Product Codes

Mustafa Cemil Coşkun

TL;DR

This paper introduces precoded polar product codes by combining non-systematic precoded polar component codes into two-dimensional product codes. It leverages SC-aimed precoding to enable efficient SCL-based component decoding and generates bit-wise soft outputs for turbo-like iterative decoding, with a focus on improving the distance spectrum. The authors show that the product-distance can be significantly increased (e.g., from 16 to 36 in representative cases) and that the proposed codes can match or surpass the performance of product codes with eBCH components and, in some cases, approach 5G LDPC performance at low rates. Numerical results on the biAWGN channel demonstrate sizable gains over competing schemes, and the method yields natural rate adaptation due to 1-bit granularity in the component codes. The work points to practical, low-latency, rate-adaptive coding approaches suitable for future standards.

Abstract

Precoded polar product codes are proposed, where selected component codes enable successive cancellation list decoding to generate bit-wise soft messages efficiently for iterative decoding while targeting optimized distance spectrum as opposed to eBCH or polar component codes. Rate compatibility is a byproduct of $1$-bit granularity in the component code design.

Precoded Polar Product Codes

TL;DR

This paper introduces precoded polar product codes by combining non-systematic precoded polar component codes into two-dimensional product codes. It leverages SC-aimed precoding to enable efficient SCL-based component decoding and generates bit-wise soft outputs for turbo-like iterative decoding, with a focus on improving the distance spectrum. The authors show that the product-distance can be significantly increased (e.g., from 16 to 36 in representative cases) and that the proposed codes can match or surpass the performance of product codes with eBCH components and, in some cases, approach 5G LDPC performance at low rates. Numerical results on the biAWGN channel demonstrate sizable gains over competing schemes, and the method yields natural rate adaptation due to 1-bit granularity in the component codes. The work points to practical, low-latency, rate-adaptive coding approaches suitable for future standards.

Abstract

Precoded polar product codes are proposed, where selected component codes enable successive cancellation list decoding to generate bit-wise soft messages efficiently for iterative decoding while targeting optimized distance spectrum as opposed to eBCH or polar component codes. Rate compatibility is a byproduct of -bit granularity in the component code design.
Paper Structure (19 sections, 1 theorem, 19 equations, 1 figure)

This paper contains 19 sections, 1 theorem, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and System Model
  4. Product Codes
  5. Code Construction
  6. Encoding
  7. Distance Spectrum
  8. Iterative (Turbo) Decoding
  9. Precoded Polar Codes
  10. Code Construction
  11. Encoding
  12. SC/SCL Decoding
  13. Non-Systematic Precoded Polar Product Codes
  14. Code Construction
  15. Encoding
  16. ...and 4 more sections

Key Result

Proposition 1

A generator matrix of the precoded polar product code, in a non-systematic form, is given by where $\boldsymbol{P}=\boldsymbol{P}_1\otimes \boldsymbol{P}_2\otimes\ldots\otimes\boldsymbol{P}_m$. If precoding matrices $\boldsymbol{P_\ell}$, $\forall\ell\in[m]$, are SC-aimed, i.e., they satisfy the properties 1) and 2) in Section sec:polar, then $\boldsymbol{P}$ is SC-aimed as well.

Figures (1)

  • Figure 1: CER vs. SNR over the biAWGN channel for the proposed codes under the described iterative decoding, where $I_{\mathrm{max}}=20$. The performance is compared to those of product codes with eBCH component codes (with BCJR for (a) and modified Chase for (c) with $20$ iterations), polar product codes and 5G NR LDPC codes (under BP decoding with $100$ iterations).

Theorems & Definitions (2)

  • Proposition 1
  • Example 1