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SCL Decoding of Non-Binary Linear Block Codes

Jingyu Lin, Li Chen, Xiaoqian Ye

TL;DR

The paper tackles the challenge of soft-decision decoding for non-binary linear block codes by proposing successive cancellation list (SCL) decoding that maps NB-LBCs over $\mathbb{F}_{2^r}$ to $r$ binary polar codewords. It establishes a one-to-$r$ binary decomposition, enabling SC/SCL decoding with complexity $\mathcal{O}(r L N \log_2 N)$ and introducing an $r$-step path sorting technique to manage path growth. The approach is validated on extended Reed-Solomon and NB-eBCH codes, where SCL decoding outperforms state-of-the-art soft-decision methods (Chase-BM and KV) in terms of finite-field arithmetic, and, for some short lengths, approaches ML decoding with moderate $L$. This work provides a practical framework for efficient soft-decision decoding of NB-LBCs, with potential impact on communications and storage systems handling bursty channels.

Abstract

Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains challenging. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F_{2^r}, where r is the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword and r binary polar codewords, SCL decoding of the r polar codes can be performed with a complexity that is sub-quadratic in the codeword length. An r-step decoding path sorting strategy is further proposed to facilitate the decoding. Simulation results on extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size.

SCL Decoding of Non-Binary Linear Block Codes

TL;DR

The paper tackles the challenge of soft-decision decoding for non-binary linear block codes by proposing successive cancellation list (SCL) decoding that maps NB-LBCs over to binary polar codewords. It establishes a one-to- binary decomposition, enabling SC/SCL decoding with complexity and introducing an -step path sorting technique to manage path growth. The approach is validated on extended Reed-Solomon and NB-eBCH codes, where SCL decoding outperforms state-of-the-art soft-decision methods (Chase-BM and KV) in terms of finite-field arithmetic, and, for some short lengths, approaches ML decoding with moderate . This work provides a practical framework for efficient soft-decision decoding of NB-LBCs, with potential impact on communications and storage systems handling bursty channels.

Abstract

Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains challenging. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F_{2^r}, where r is the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword and r binary polar codewords, SCL decoding of the r polar codes can be performed with a complexity that is sub-quadratic in the codeword length. An r-step decoding path sorting strategy is further proposed to facilitate the decoding. Simulation results on extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size.

Paper Structure

This paper contains 9 sections, 1 theorem, 27 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar Codes
  4. Transformation From B-LBCs To Polar Codes
  5. Decomposition of NB-LBCs
  6. SCL Decoding of NBLBCs
  7. SC and SCL Decoding
  8. $r$-Step Path Sorting
  9. Simulation Results

Key Result

Theorem 1

Let $\mathcal{C}_{\rm NB}$ be an $(N=2^n, K)$ NB-LBC defined over $\mathbb{F}_{2^r}$. The binary composition of any codeword $\bm{c} \in \mathcal{C}_{\rm NB}$ can be represented as the concatenation of $r$ permuted binary polar codewords.

Figures (4)

  • Figure 1: Block diagram of the $r$-step path sorting ($L = 4$).
  • Figure 2: SCL decoding performance of the $(32, 15)$ eRS code.
  • Figure 3: SCL decoding performance of the $(64, 27)$ NB-eBCH code over $\mathbb{F}_{4}$.
  • Figure 4: SCL decoding performance of the $(16, 7)$ eRS code.

Theorems & Definitions (2)

  • Theorem 1
  • Remark 1