1,393 research outputs found
In-depth comparison of the Berlekamp--Massey--Sakata and the Scalar-FGLM algorithms: the adaptive variants
The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both
compute the ideal of relations of a multidimensional linear recurrent
sequence.Whenever quering a single sequence element is prohibitive, the
bottleneck of these algorithms becomes the computation of all the needed
sequence terms. As such, having adaptive variants of these algorithms, reducing
the number of sequence queries, becomes mandatory.A native adaptive variant of
the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive
Scalar-FGLM algorithm.In this paper, our first contribution is to make the
Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to
avoid some useless relation test-ings. This variant allows us to divide by four
in dimension 2 and by seven in dimension 3 the number of basic operations
performed on some sequence family.Then, we compare the two adaptive algorithms.
We show that their behaviors differ in a way that it is not possible to tweak
one of the algorithms in order to mimic exactly the behavior of the other. We
detail precisely the differences and the similarities of both algorithms and
conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries
and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata
algorithm.We also show that these variants are always more efficient than the
original algorithms
Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations
Sparse polynomial interpolation, sparse linear system solving or modular
rational reconstruction are fundamental problems in Computer Algebra. They come
down to computing linear recurrence relations of a sequence with the
Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial
interpolation and multidimensional cyclic code decoding require guessing linear
recurrence relations of a multivariate sequence.Several algorithms solve this
problem. The so-called Berlekamp-Massey-Sakata algorithm (1988) uses polynomial
additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on
linear algebra operations on a multi-Hankel matrix, a multivariate
generalization of a Hankel matrix. The Artinian Gorenstein border basis
algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for
computing the Gr{\"o}bner basis of the ideal of relations of a sequence based
solely on multivariate polynomial arithmetic. This algorithm allows us to both
revisit the Berlekamp-Massey-Sakata algorithm through the use of polynomial
divisions and to completely revise the Scalar-FGLM algorithm without linear
algebra operations.A key observation in the design of this algorithm is to work
on the mirror of the truncated generating series allowing us to use polynomial
arithmetic modulo a monomial ideal. It appears to have some similarities with
Pad{\'e} approximants of this mirror polynomial.As an addition from the paper
published at the ISSAC conferance, we give an adaptive variant of this
algorithm taking into account the shape of the final Gr{\"o}bner basis
gradually as it is discovered. The main advantage of this algorithm is that its
complexity in terms of operations and sequence queries only depends on the
output Gr{\"o}bner basis.All these algorithms have been implemented in Maple
and we report on our comparisons
A Review :Implementation of Reed Solomon Error Correction & Detec-tion For Wireless Network 802.16
The reed Solomon (255,239) are error-correcting & detecting code. Reed-Solomon codes are the most frequently used digital error control. It is also called as forword error code. The main part of reed-Solomon encoder is the linear feedback shift register that is implemented using VHDL A pipelined RS decoders is proposed of reducing the hardware complexity use the pipelined GFmultiplier in the syndrome computation block, KES block, Forney block, Chien search block and error correction block for provides low com-plexity the extended inversion less Massey-Berlekamp algorithm is used. The extended inversion less Massey-Berlekamp algorithm overcomes both the error locator polynomial and the error evaluator polynomial at the same time
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