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Successive Cancellation List Decoding of Extended Reed-Solomon Codes

Xiaoqian Ye, Jingyu Lin, Junjie Huang, Li Chen, Chang-An Zhao

TL;DR

The paper tackles soft-decision decoding of extended Reed-Solomon codes over $\,\mathbb{F}_{2^n}\,$ by transforming each eRS codeword into $n$ binary polar codes and applying SC/SCL decoding via a pre-transformed matrix $\mathbf{M}$. It provides a theoretical analysis of the matrix, proving a linearly independent column property in a key index set $\mathcal{D}$ and deriving a lower bound on SC performance through channel degradation, with a constructive discussion of information/frozen symbol distribution. Numerical results for length-32 eRS codes show that SCL decoding can outperform KV and Chase-BM in certain rate regimes, highlighting practical gains for short-to-medium length codes. The work demonstrates a viable framework that leverages polar decoding for non-binary eRS codes, offering insights into how pre-transformed matrix design influences decoding effectiveness and complexity.

Abstract

Reed-Solomon (RS) codes are an important class of non-binary error-correction codes. They are particularly competent in correcting burst errors, being widely applied in modern communications and data storage systems. This also thanks to their distance property of reaching the Singleton bound, being the maximum distance separable (MDS) codes. This paper proposes a new list decoding for extended RS (eRS) codes defined over a finite field of characteristic two, i.e., F_{2^n}. It is developed based on transforming an eRS code into n binary polar codes. Consequently, it can be decoded by the successive cancellation (SC) decoding and further their list decoding, i.e., the SCL decoding. A pre-transformed matrix is required for reinterpretating the eRS codes, which also determines their SC and SCL decoding performances. Its column linear independence property is studied, leading to theoretical characterization of their SC decoding performance. Our proposed decoding and analysis are validated numerically.

Successive Cancellation List Decoding of Extended Reed-Solomon Codes

TL;DR

The paper tackles soft-decision decoding of extended Reed-Solomon codes over by transforming each eRS codeword into binary polar codes and applying SC/SCL decoding via a pre-transformed matrix . It provides a theoretical analysis of the matrix, proving a linearly independent column property in a key index set and deriving a lower bound on SC performance through channel degradation, with a constructive discussion of information/frozen symbol distribution. Numerical results for length-32 eRS codes show that SCL decoding can outperform KV and Chase-BM in certain rate regimes, highlighting practical gains for short-to-medium length codes. The work demonstrates a viable framework that leverages polar decoding for non-binary eRS codes, offering insights into how pre-transformed matrix design influences decoding effectiveness and complexity.

Abstract

Reed-Solomon (RS) codes are an important class of non-binary error-correction codes. They are particularly competent in correcting burst errors, being widely applied in modern communications and data storage systems. This also thanks to their distance property of reaching the Singleton bound, being the maximum distance separable (MDS) codes. This paper proposes a new list decoding for extended RS (eRS) codes defined over a finite field of characteristic two, i.e., F_{2^n}. It is developed based on transforming an eRS code into n binary polar codes. Consequently, it can be decoded by the successive cancellation (SC) decoding and further their list decoding, i.e., the SCL decoding. A pre-transformed matrix is required for reinterpretating the eRS codes, which also determines their SC and SCL decoding performances. Its column linear independence property is studied, leading to theoretical characterization of their SC decoding performance. Our proposed decoding and analysis are validated numerically.
Paper Structure (11 sections, 3 theorems, 36 equations, 2 figures, 2 tables)

This paper contains 11 sections, 3 theorems, 36 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. eRS Codes
  4. Polar Codes
  5. SCL Decoding of eRS Codes
  6. Transformation from eRS Codes to Polar Codes
  7. SC and SCL Decoding
  8. SC Decoding Performance Analysis
  9. Linearly Independent Columns of Pre-transformed Matrix
  10. SC Decoding Performance Lower Bound
  11. Numerical study

Key Result

Lemma 1

Given an $(N, K)$ eRS code for all $\bm{{\rm P}}\in\mathbb{F}_2^{N\times N}$, columns of $\mathbf{M}^{\mathcal{D}}$ are linearly independent.

Figures (2)

  • Figure 1: Transformation from an eRS codeword into $n$ permuted binary polar codewords.
  • Figure 2: SCL Decoding Performance of Length-32 eRS Codes.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 3