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