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Fast Successive-Cancellation Decoding of 2 x 2 Kernel Non-Binary Polar Codes: Identification, Decoding and Simplification

Ali Farsiabi, Hamid Ebrahimzad, Masoud Ardakani, Chuandong Li

TL;DR

This work tackles the latency bottleneck of successive-cancellation decoding for non-binary polar codes (NBPCs) built from $2\times 2$ kernels. It introduces a fast LLR-domain NBSC (LNBSC) approach by identifying a rich set of non-binary special nodes and deriving low-complexity, parallel decoding procedures for each, together with a hardware-friendly NBPC structure. The paper also proposes a simplified NBPC design that minimizes permutation and kernel-parameter overhead while preserving performance. Empirical results show substantial latency reductions (up to $95\%$) with negligible degradation in error-rate performance, enabling practical, low-latency NBPC implementations.

Abstract

Non-binary polar codes (NBPCs) decoded by successive cancellation (SC) algorithm have remarkable bit-error-rate performance compared to the binary polar codes (BPCs). Due to the serial nature, SC decoding suffers from large latency. The latency issue in BPCs has been the topic of extensive research and it has been notably resolved by the introduction of fast SC-based decoders. However, the vast majority of research on NBPCs is devoted to issues concerning design and efficient implementation. In this paper, we propose fast SC decoding for NBPCs constructed based on 2 x 2 kernels. In particular, we identify various non-binary special nodes in the SC decoding tree of NBPCs and propose their fast decoding. This way, we avoid traversing the full decoding tree and significantly reduce the decoding delay compared to symbol-by-symbol SC decoding. We also propose a simplified NBPC structure that facilitates the procedure of non-binary fast SC decoding. Using our proposed fast non-binary decoder, we observed an improvement of up to 95% in latency concerning the original SC decoding. This is while our proposed fast SC decoder for NBPCs incurs no error-rate loss.

Fast Successive-Cancellation Decoding of 2 x 2 Kernel Non-Binary Polar Codes: Identification, Decoding and Simplification

TL;DR

This work tackles the latency bottleneck of successive-cancellation decoding for non-binary polar codes (NBPCs) built from kernels. It introduces a fast LLR-domain NBSC (LNBSC) approach by identifying a rich set of non-binary special nodes and deriving low-complexity, parallel decoding procedures for each, together with a hardware-friendly NBPC structure. The paper also proposes a simplified NBPC design that minimizes permutation and kernel-parameter overhead while preserving performance. Empirical results show substantial latency reductions (up to ) with negligible degradation in error-rate performance, enabling practical, low-latency NBPC implementations.

Abstract

Non-binary polar codes (NBPCs) decoded by successive cancellation (SC) algorithm have remarkable bit-error-rate performance compared to the binary polar codes (BPCs). Due to the serial nature, SC decoding suffers from large latency. The latency issue in BPCs has been the topic of extensive research and it has been notably resolved by the introduction of fast SC-based decoders. However, the vast majority of research on NBPCs is devoted to issues concerning design and efficient implementation. In this paper, we propose fast SC decoding for NBPCs constructed based on 2 x 2 kernels. In particular, we identify various non-binary special nodes in the SC decoding tree of NBPCs and propose their fast decoding. This way, we avoid traversing the full decoding tree and significantly reduce the decoding delay compared to symbol-by-symbol SC decoding. We also propose a simplified NBPC structure that facilitates the procedure of non-binary fast SC decoding. Using our proposed fast non-binary decoder, we observed an improvement of up to 95% in latency concerning the original SC decoding. This is while our proposed fast SC decoder for NBPCs incurs no error-rate loss.
Paper Structure (22 sections, 13 theorems, 61 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 22 sections, 13 theorems, 61 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. Non-binary polar codes
  5. SC Decoding of NBPC
  6. Proposed Fast-LNBSC Decoding Based on Special Nodes
  7. Rate-0 Node
  8. Rate-1 Node
  9. M-REP Node
  10. M-SPC Node
  11. Type-I Node
  12. Type-II Node
  13. Type-III Node
  14. Type-IV Node
  15. Type-V Node
  16. ...and 7 more sections

Key Result

Lemma 1

Consider $\mathbf{G}^{\bigotimes k}_2 = [\mathbf{g}^{(k)}_0, \dots, \mathbf{g}^{(k)}_{2^k-1}]$, $k \in \mathbb{Z^+}$, where $\mathbf{g}^{(k)}_{i}=[{g}^{(k)}_{i,0}\dots,{g}^{(k)}_{i,2^k-1}]^T$, $i=0,\dots, 2^k-1$, are column-vectors with $2^k$ elements that form the columns of matrix $\mathbf{G}^{\bi where $\mathbf{g}^{(k)^{-1}}_0$ is the first column of $\mathbf{G}^{{\bigotimes k}^{-1}}_2$, i.e.,

Figures (9)

  • Figure 1: Non-binary $2 \times 2$ kernel with $\mu, \gamma, \delta \in \mathbb{GF}(q) \backslash 0$.
  • Figure 2: Binary-tree representation of a non-binary polar code with $N=16$ and $K=8$. The black and white leaf nodes are information and frozen symbols, respectively.
  • Figure 3: Three types of messages that are calculated at node $( \nu, s)$: (a) CN operation (message toward the left child), (b) VN operation (message toward the right child), and (c) message toward the parent node.
  • Figure 4: General structures of (a) GM-REP and (b) GM-PC nodes.
  • Figure 5: Example of the general structure of NBPCs with variable kernel-coefficients in polarization units. Each gray box corresponds to a node at level $s$ of the binary-tree of the polar code.
  • ...and 4 more figures

Theorems & Definitions (36)

  • Remark 1
  • Example 1
  • Definition 1
  • Example 2
  • Remark 2
  • Definition 2
  • Example 3
  • Definition 3
  • Lemma 1
  • Example 4
  • ...and 26 more