572 research outputs found
Review of finite fields: Applications to discrete Fourier, transforms and Reed-Solomon coding
An attempt is made to provide a step-by-step approach to the subject of finite fields. Rigorous proofs and highly theoretical materials are avoided. The simple concepts of groups, rings, and fields are discussed and developed more or less heuristically. Examples are used liberally to illustrate the meaning of definitions and theories. Applications include discrete Fourier transforms and Reed-Solomon coding
Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes
In this paper, we present a new basis of polynomial over finite fields of
characteristic two and then apply it to the encoding/decoding of Reed-Solomon
erasure codes. The proposed polynomial basis allows that -point polynomial
evaluation can be computed in finite field operations with
small leading constant. As compared with the canonical polynomial basis, the
proposed basis improves the arithmetic complexity of addition, multiplication,
and the determination of polynomial degree from
to . Based on this basis, we then develop the encoding and
erasure decoding algorithms for the Reed-Solomon codes. Thanks to
the efficiency of transform based on the polynomial basis, the encoding can be
completed in finite field operations, and the erasure decoding
in finite field operations. To the best of our knowledge, this
is the first approach supporting Reed-Solomon erasure codes over
characteristic-2 finite fields while achieving a complexity of ,
in both additive and multiplicative complexities. As the complexity leading
factor is small, the algorithms are advantageous in practical applications
New Decoding of Reed-Solomon Codes Based on FFT and Modular Approach
Decoding algorithms for Reed--Solomon (RS) codes are of great interest for
both practical and theoretical reasons. In this paper, an efficient algorithm,
called the modular approach (MA), is devised for solving the Welch--Berlekamp
(WB) key equation. By taking the MA as the key equation solver, we propose a
new decoding algorithm for systematic RS codes. For RS codes, where
is the code length and is the code dimension, the proposed decoding
algorithm has both the best asymptotic computational complexity and the smallest constant factor achieved to date. By
comparing the number of field operations required, we show that when decoding
practical RS codes, the new algorithm is significantly superior to the existing
methods in terms of computational complexity. When decoding the
RS code defined over , the new algorithm is 10 times
faster than a conventional syndrome-based method. Furthermore, the new
algorithm has a regular architecture and is thus suitable for hardware
implementation
A common operator for FFT and FEC decoding
International audienceIn the Software Radio context, the parametrization is becoming an important topic especially when it comes to multistandard designs. This paper capitalizes on the Common Operator technique to present new common structures for the FFT and FEC decoding algorithms. A key benefit of exhibiting common operators is the regular architecture it brings when implemented in a Common Operator Bank (COB). This regularity makes the architecture open to future function mapping and adapted to accommodated silicon technology variability through dependable design
Fast Fourier transform via automorphism groups of rational function fields
The Fast Fourier Transform (FFT) over a finite field computes
evaluations of a given polynomial of degree less than at a specifically
chosen set of distinct evaluation points in . If or
is a smooth number, then the divide-and-conquer approach leads to the fastest
known FFT algorithms. Depending on the type of group that the set of evaluation
points forms, these algorithms are classified as multiplicative (Math of Comp.
1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a
unified framework for FFT algorithms that include both multiplicative and
additive FFT algorithms as special cases, and beyond: our framework also works
when is smooth, while all known results require or to be
smooth. For the new case where is smooth (this new case was not
considered before in literature as far as we know), we show that if is a
divisor of that is -smooth for a real , then our FFT needs
arithmetic operations in . Our unified framework is
a natural consequence of introducing the algebraic function fields into the
study of FFT
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