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

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