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Reed-Solomon Codes over Cyclic Polynomial Ring with Lower Encoding/Decoding Complexity

Wenhao Liu, Zhengyi Jiang, Zhongyi Huang, Linqi Song, Hanxu Hou

TL;DR

The paper addresses high XOR-based encoding/decoding complexity of Reed-Solomon codes by constructing RS codes over the cyclic polynomial ring $\mathbb{F}_2[x]/(M_p(x))$ and developing FFT and modular techniques on this ring. It presents a CRT-based construction, proves MDS properties, and extends FFT/IFFT along with FDMA-based decoding to the ring, achieving reduced complexity. Empirically, the ring-based approach yields 17.9% encoding and 7.5% decoding complexity reductions for a $(2048,1984)$ code compared to RS codes over a finite field. This work enables lower-complexity RS coding for large-length codes and offers practical pathways for fast encoding/decoding in storage and communications.

Abstract

Reed-Solomon (RS) codes are constructed over a finite field that have been widely employed in storage and communication systems. Many fast encoding/decoding algorithms such as fast Fourier transform (FFT) and modular approach are designed for RS codes to reduce the encoding/decoding complexity defined as the number of XORs involved in the encoding/decoding procedure. In this paper, we present the construction of RS codes over the cyclic polynomial ring $ \mathbb{F}_2[x]/(1+x+\ldots+x^{p-1})$ and show that our codes are maximum distance separable (MDS) codes. Moreover, we propose the FFT and modular approach over the ring that can be employed in our codes for encoding/decoding complexity reduction. We show that our codes have 17.9\% encoding complexity reduction and 7.5\% decoding complexity reduction compared with RS codes over finite field, for $(n,k)=(2048,1984)$.

Reed-Solomon Codes over Cyclic Polynomial Ring with Lower Encoding/Decoding Complexity

TL;DR

The paper addresses high XOR-based encoding/decoding complexity of Reed-Solomon codes by constructing RS codes over the cyclic polynomial ring and developing FFT and modular techniques on this ring. It presents a CRT-based construction, proves MDS properties, and extends FFT/IFFT along with FDMA-based decoding to the ring, achieving reduced complexity. Empirically, the ring-based approach yields 17.9% encoding and 7.5% decoding complexity reductions for a code compared to RS codes over a finite field. This work enables lower-complexity RS coding for large-length codes and offers practical pathways for fast encoding/decoding in storage and communications.

Abstract

Reed-Solomon (RS) codes are constructed over a finite field that have been widely employed in storage and communication systems. Many fast encoding/decoding algorithms such as fast Fourier transform (FFT) and modular approach are designed for RS codes to reduce the encoding/decoding complexity defined as the number of XORs involved in the encoding/decoding procedure. In this paper, we present the construction of RS codes over the cyclic polynomial ring and show that our codes are maximum distance separable (MDS) codes. Moreover, we propose the FFT and modular approach over the ring that can be employed in our codes for encoding/decoding complexity reduction. We show that our codes have 17.9\% encoding complexity reduction and 7.5\% decoding complexity reduction compared with RS codes over finite field, for .
Paper Structure (18 sections, 7 theorems, 28 equations, 2 tables, 3 algorithms)

This paper contains 18 sections, 7 theorems, 28 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. RS Codes over Cyclic Polynomial Ring
  3. Construction
  4. Encoding and Decoding Methods
  5. FFT and IFFT Algorithms over $\{\mathbb{F}_2[x]/(p_i(x))\}_{i\in[t]}$ and $\mathcal{R}_p$
  6. Encoding Algorithm
  7. Decoding Algorithm
  8. Syndrome
  9. Solving the Key Equation by FDMA
  10. Chein search
  11. Formal Derivative
  12. Forney's Formula
  13. Reduce Encoding/decoding complexity
  14. Efficient Multiplication by Circular Shifts
  15. Preprocessing in the Ring $\mathbb{F}_2[x]/(1+x^p)$
  16. ...and 3 more sections

Key Result

Lemma 1

The polynomial $r(x)\in \mathcal{R}_p$ is invertible if and only if $\forall i \in [t],r(x)\not\equiv0\pmod{p_i(x)}$.

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 2
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8