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ORCAS Codes: A Flexible Generalization of Polar Codes with Low-Complexity Decoding

Andreas Zunker, Marvin Rübenacke, Stephan ten Brink

TL;DR

This work considers the recursive Plotkin concatenation of optimal low-rate and high-rate codes based on simplex codes and their duals to achieve a performance that is at least as good as that of polar codes.

Abstract

Motivated by the need for channel codes with low-complexity soft-decision decoding algorithms, we consider the recursive Plotkin concatenation of optimal low-rate and high-rate codes based on simplex codes and their duals. These component codes come with low-complexity maximum likelihood (ML) decoding which, in turn, enables efficient successive cancellation (SC)-based decoding. As a result, the proposed optimally recursively concatenated simplex (ORCAS) codes achieve a performance that is at least as good as that of polar codes. For practical parameters, the proposed construction significantly outperforms polar codes in terms of block error rate by up to 0.5 dB while maintaining similar decoding complexity. Furthermore, the codes offer greater flexibility in codeword length than conventional polar codes.

ORCAS Codes: A Flexible Generalization of Polar Codes with Low-Complexity Decoding

TL;DR

This work considers the recursive Plotkin concatenation of optimal low-rate and high-rate codes based on simplex codes and their duals to achieve a performance that is at least as good as that of polar codes.

Abstract

Motivated by the need for channel codes with low-complexity soft-decision decoding algorithms, we consider the recursive Plotkin concatenation of optimal low-rate and high-rate codes based on simplex codes and their duals. These component codes come with low-complexity maximum likelihood (ML) decoding which, in turn, enables efficient successive cancellation (SC)-based decoding. As a result, the proposed optimally recursively concatenated simplex (ORCAS) codes achieve a performance that is at least as good as that of polar codes. For practical parameters, the proposed construction significantly outperforms polar codes in terms of block error rate by up to 0.5 dB while maintaining similar decoding complexity. Furthermore, the codes offer greater flexibility in codeword length than conventional polar codes.

Paper Structure

This paper contains 26 sections, 5 theorems, 32 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Binary Linear Block Codes
  5. Plotkin Construction, Successive Cancellation Decoding, and Polar Codes
  6. Density Evolution / Gaussian Approximation
  7. Error Probability of Binary Linear Block Codes
  8. Optimality of Low-Rate and High-Rate Codes
  9. Near Optimal Low-Rate Codes: Naturally Punctured Repeated Simplex Codes
  10. Construction and Optimality
  11. Weight Distribution
  12. Optimal High-Rate Codes: NPRS Dual Codes
  13. Construction and Optimality
  14. Weight Distribution
  15. Design
  16. ...and 11 more sections

Key Result

Theorem 1

An NPRS code of length $n$ is distance-optimal for dimensions $k \in \kappa_n(l_\mathrm{b}(n))$, where and $\mathcal{A}(n,k) = \{0\}\cup\mleft\{2^{i}-i,\dots,2^{i}\mright\}$, $i=l_\mathrm{b}(a(n,k)-1)$.

Figures (2)

  • Figure 1: Top: Factor graph of a $(96,48,8)$ORCAS code with the $(n,k,d)$ parameters of the component codes. Bottom: Factor graphs of the component decoders. Squares and circles denote check and variable nodes, respectively.
  • Figure 2: BLER performance comparison of ORCAS codes and polar codes under SC decoding. All codes are optimized for a BLER of $10^{-6}$ by DE.

Theorems & Definitions (7)

  • Definition 1: Low-rate code
  • Definition 2: High-rate code
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Corollary 1
  • Theorem 3