A method for computation of the discrete Fourier transform over a finite field

Fedorenko, Sergei V.

doi:10.1134/s0032946006020074

Cited by 32 publications

(83 citation statements)

References 4 publications

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“…Since the multiplication between a cyclic matrix and a vector can be done by efficient bilinear algorithm of cyclic convolution, CFFTs can be computed by F = AQ(c · Pf ), where Q and P are binary matrices, c is a pre-computed vector, and · denotes an entry-wise multiplication between two vectors. Two variants of DCFFTs, referred to as inverse CFFTs (ICFFTs) [6] and symmetric CFFTs (SCFFTs) [9], respectively, compute the DFTs by…”

Section: Background a Cffts And Ccfts Over Finite Fieldsmentioning

confidence: 99%

“…CFFTs proposed in [6], [8], [9] have low multiplicative complexities, but they have very high additive complexities. By using techniques such as the common subexpression elimination (CSE) algorithm in [10], the additive complexities of CFFTs can be significantly reduced, leading to small overall computational complexities for DFTs with lengths up to 1024 [10].…”

Section: Introductionmentioning

confidence: 99%

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Reduced-Complexity Decoders of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

Yan

Lin

2012

IEEE Trans. Signal Process.

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Abstract-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 (2720, 2550) shortened RS code over GF(2 12 ). 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. Fast algorithms for discrete Fourier transforms (DFTs) over finite fields are promising techniques to overcome this obstacle. This is because all steps except the key equation solver in syndrome-based hard-decision RS decoders [1] -syndrome computation, Chien search, and error magnitude evaluationare polynomial evaluations. Hence, they can be formulated as DFTs over finite fields.Recently, cyclotomic fast Fourier transforms (CFFTs) over finite fields have been used to reduce the complexities of RS decoders [6], [7]. CFFTs proposed in [6], [8], [9] have low multiplicative complexities, but they have very high additive complexities. By using techniques such as the common subexpression elimination (CSE) algorithm in [10], the additive

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Section: Background a Cffts And Ccfts Over Finite Fieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduced-Complexity Decoders of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

Yan

Lin

2012

IEEE Trans. Signal Process.

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“…Using an approach similar to those in previous works (see, for example, [5]), cyclotomic FFT (CFFT) was recently proposed [6] and two variants were subsequently considered [7], [8]. To avoid confusion, we refer to the CFFT proposed in [6] as direct CFFT (DCFFT) and those in [7] and [8] henceforth in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…To avoid confusion, we refer to the CFFT proposed in [6] as direct CFFT (DCFFT) and those in [7] and [8] henceforth in this paper. DCFFT has been shown to be efficient for full DFTs of lengths up to 511 [6], and ICFFT and SCFFT are particularly suitable for partial DFTs, which compute only part of the spectral components and are important for such operations as syndrome computation of Reed-Solomon decoders [7], [8].…”

Section: Introductionmentioning

confidence: 99%

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Cyclotomic FFTs With Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm

Chen

Yan

2009

IEEE Trans. Signal Process.

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Abstract-In this paper, we first propose a novel common subexpression elimination (CSE) algorithm for matrix-vector multiplications over characteristic-2 fields. As opposed to previously proposed CSE algorithms, which usually focus on complexity savings due to recurrences of subexpressions, our CSE algorithm achieves two types of complexity reductions, differential savings and recurrence savings, by taking advantage of the cancelation property of characteristic-2 fields. Using our CSE algorithm, we reduce the additive complexities of cyclotomic fast Fourier transforms (CFFTs). Using a weighted sum of the numbers of multiplications and additions as a metric, our CFFTs achieve smaller total complexities than previously proposed CFFTs and other FFTs, requiring both fewer multiplications and fewer additions in many cases.

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Representation of Cascade Codes in the Frequency Domain

Кuznetsov

Pushkar'ov

Serhiienko

et al. 2020

Data-Centric Business and Applications

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The mathematical apparatus of the multidimensional discrete Fourier transform over finite fields is considered. Methods for the description of linear block codes in the frequency domain are investigated. It is shown that, in contrast to iterative codes (code-products), cascade codes in the general case cannot be described in the frequency domain in terms of multidimensional spectra. Analytic expressions are obtained that establish a one-to-one functional correspondence between the spectrum of a sequence over a finite field and the spectra of the corresponding words obtained by limiting this word to a subfield. A general solution of the problem of representation of cascade codes in the frequency domain is obtained, which allows constructing in the frequency domain using computationally efficient algorithms of encoding and decoding, and the derived analytic dependences of components of multidimensional spectra.

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A method for computation of the discrete Fourier transform over a finite field

Cited by 32 publications

References 4 publications

Reduced-Complexity Decoders of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

Reduced-Complexity Decoders of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

Cyclotomic FFTs With Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm

Representation of Cascade Codes in the Frequency Domain

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