3 research outputs found
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
Reduced-Complexity Decoder of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms
Long Reed-Solomon (RS) codes are desirable for digital communication and
storage systems due to their improved error performance, but the high
computational complexity of their decoders is a key obstacle to their adoption
in practice. As discrete Fourier transforms (DFTs) can evaluate a polynomial at
multiple points, efficient DFT algorithms are promising in reducing the
computational complexities of syndrome based decoders for long RS codes. In
this paper, we first propose partial composite cyclotomic Fourier transforms
(CCFTs) and then devise syndrome based decoders for long RS codes over large
finite fields based on partial CCFTs. The new decoders based on partial CCFTs
achieve a significant saving of computational complexities for long RS codes.
Since partial CCFTs have modular and regular structures, the new decoders are
suitable for hardware implementations. To further verify and demonstrate the
advantages of partial CCFTs, we implement in hardware the syndrome computation
block for a shortened RS code over GF. In comparison
to previous results based on Horner's rule, our hardware implementation not
only has a smaller gate count, but also achieves much higher throughputs.Comment: 7 pages, 1 figur
A novel method for computation of the discrete Fourier transform over characteristic two finite field of even extension degree
A novel method for computation of the discrete Fourier transform over a
finite field with reduced multiplicative complexity is described. If the number
of multiplications is to be minimized, then the novel method for the finite
field of even extension degree is the best known method of the discrete Fourier
transform computation. A constructive method of constructing for a cyclic
convolution over a finite field is introduced.Comment: 35 pages. Submitted to IEEE Transactions on Information Theor