Efficient Decoding of a Class of Reed-Solomon Codes Over Fermat Fields

In this paper, we present an efficient decoding algorithm for a class of Reed-Solomon (RS) codes over Fermat field
$\mathbb{F}_{2^{r}+1}$
. We show that the Fermat number transform can be used to speed up the syndrome computation and the Chien search. The implementation architectures are designed for the two blocks. The key equation is then derived. When using the RS code in practice, there arises the issue that a
$(2^{r}+1)$
-ary symbol is less efficiently represented by a tuple of
$(r+1)$
bits. We present a nested coding scheme based on RS code and single parity-check (SPC) code to harness the inefficiency. A modified Wagner algorithm is proposed for decoding the inner (nonlinear) code and is proved to be an ML decoding over the BPSK-modulated AWGN channel. Simulation results show that the proposed RS-SPC nested coding scheme yields a considerable performance gain compared to the stand-alone RS coding scheme.