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

TL;DR

This paper introduces Reed-Solomon codes over a cyclic polynomial ring, achieving lower encoding and decoding complexity compared to traditional finite field-based RS codes, with practical reductions demonstrated.

Contribution

It constructs MDS Reed-Solomon codes over a cyclic polynomial ring and develops FFT and modular algorithms for reduced complexity encoding and decoding.

Findings

17.9% encoding complexity reduction

7.5% decoding complexity reduction

Effective for (n,k)=(2048,1984)

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 $(n,k)=(2048,1984)$.