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Efficient LLR-Domain Decoding of ABS+ Polar Codes

Mikhail Chernikov, Peter Trifonov

TL;DR

An LLR-domain implementation of the SCL decoder of ABS+ polar codes is presented and the SCL algorithm is optimized in order to reduce the complexity requirements for the LLRs computation.

Abstract

ABS+ polar codes are a generalization of Arikan polar codes that provides much faster polarization. We present an LLR-domain implementation of the SCL decoder of ABS+ polar codes. Furthermore, we optimize the SCL algorithm in order to reduce the complexity requirements for the LLRs computation. In comparison with classical polar codes, the proposed approach requires less number of arithmetic operations in the SCL decoder to obtain the fixed frame error rate (FER) at high-SNR region.

Efficient LLR-Domain Decoding of ABS+ Polar Codes

TL;DR

An LLR-domain implementation of the SCL decoder of ABS+ polar codes is presented and the SCL algorithm is optimized in order to reduce the complexity requirements for the LLRs computation.

Abstract

ABS+ polar codes are a generalization of Arikan polar codes that provides much faster polarization. We present an LLR-domain implementation of the SCL decoder of ABS+ polar codes. Furthermore, we optimize the SCL algorithm in order to reduce the complexity requirements for the LLRs computation. In comparison with classical polar codes, the proposed approach requires less number of arithmetic operations in the SCL decoder to obtain the fixed frame error rate (FER) at high-SNR region.
Paper Structure (16 sections, 3 theorems, 43 equations, 6 figures, 5 algorithms)

This paper contains 16 sections, 3 theorems, 43 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Classical polar codes
  4. Successive cancellation decoding
  5. Successive cancellation list decoding
  6. ABS+ polar codes
  7. SC decoding of ABS+ polar codes in the probability domain
  8. LLR-Domain Computations For Decoding of ABS+ Polar Codes
  9. Evolution of the approximate LLRs
  10. Reusing intermediate values
  11. Omitting calculations of unused LLRs
  12. The proposed LLR-based SC decoding algorithm
  13. SCL decoding in the LLR domain
  14. Numeric Results
  15. Conclusions
  16. ...and 1 more sections

Key Result

Lemma 1

Let us fix the matrices $Q^\text{ABS+}_{1}, Q^\text{ABS+}_{2}, \dots, Q^\text{ABS+}_{m},\ m \ge 2$ according to eq:qabs-def with the sets $\mathcal{I}_S^{(m)}$ and $\mathcal{I}_A^{(m)}$ satisfying the conditions (item:cond-1)-(item:cond-2) from Section subsect:abs-polar-codes. For every $i:\ 1 \leq

Figures (6)

  • Figure 1: Polarization of a channel $W: \{0,1\} \to \mathcal{Y}$.
  • Figure 2: Circuit for the $(8,4)$ ABS+ polar code with $\mathcal{F} = \{1,2,3,4\}$, $\mathcal{I}_S^{(3)} = \{4\},\ \mathcal{I}_A^{(3)} = \emptyset,\ \mathcal{I}_S^{(2)} = \emptyset,\ \mathcal{I}_A^{(2)} = \{2\}$. Additional transforms of adjacent bits are highlighted.
  • Figure 3: The tree of the recursion in the SC algorithm applied to an ABS+ polar code with $\mathcal{I}_S^{(2)} = \emptyset,\ \mathcal{I}_A^{(2)} = \{1\},\ \mathcal{I}_S^{(3)} = \{4\},\ \mathcal{I}_A^{(3)} = \emptyset$.
  • Figure 4: The tree of the recursion in the proposed SC algorithm applied to an ABS+ polar code with $\mathcal{I}_S^{(2)} = \emptyset,\ \mathcal{I}_A^{(2)} = \{1\},\ \mathcal{I}_S^{(3)} = \{4\},\ \mathcal{I}_A^{(3)} = \emptyset$. The algorithm pefrorms depth-first-search starting from the root node $(0,1)$; children are being processed from left to right. We use a black edge from $(\lambda,i)$ to $(\lambda+1,i')$ if $2i \not\in \mathcal{I}^{(\lambda+1)}$. Otherwise the edge is red.
  • Figure 5: Comparison of $(1024,512)$ polar codes.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1