Cyclotomic FFT of Length 2047 Based on a Novel 11-point Cyclic Convolution

Discrete Fourier transforms (DFTs) over finite fields have widespread applications in error correction coding [1]. For Reed-Solomon (RS) codes, all syndrome-based bounded distance decoding methods involve DFTs over finite fields [1]: syndrome computation and the Chien search are both evaluations of polynomials and hence can be viewed as DFTs; inverse DFTs are used to recover transmitted codewords in transform-domain decoders. Thus efficient DFT algorithms can be used to reduce the complexity of RS decoders. For example, using the prime-factor fast Fourier transform (FFT) in [2], Truong et al. proposed [3] an inverse-free transform-domain RS decoder with substantially lower complexity than time-domain decoders; FFT techniques are used to compute syndromes for time-domain decoders in [4]. Cyclotomic FFT was proposed recently in [5] and two variations were subsequently considered in [6], [7]. Compared with other FFT techniques [2], [8], CFFTs in [5]–[7] achieve significantly lower multiplicative complexities, which makes them very attractive. But their additive complexities (numbers of additions required) are very high if implemented directly. A common subexpression elimination (CSE) algorithm was proposed to significantly reduce the additive complexities of CFFTs in [9]. Along with those full CFFTs, reduced-complexity partial and dual partial CFFTs were used to design low complexity RS decoders in [10]. The lengths of CFFTs in [9] are only up to 1023 while longer CFFTs are required to decode long RS codes. To pursuit a length-2047 CFFT, 11-point cyclic convolution over characteristic-2 fields is necessary, which is not readily available in the literature. In this manuscript, we first propose a novel 11-point cyclic convolution for characteristic-2 fields in Section II. Based on this cyclic convolution, a length-2047 CFFT is presented in Section III. Using the same approach, CFFTs of any lengths that divide 2047 can also be constructed.