On Sequences, Rational Functions and Decomposition

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences.Comment: Several more typos corrected. To appear in J. Applied Algebra in Engineering, Communication and Computing. The final publication version is available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-

Minimal Polynomial Algorithms for Finite Sequences

We show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of a recent algorithm of A. Salagean.Comment: Section 2 added, remarks and references expanded. To appear in IEEE Transactions on Information Theory

On the key equation for n-dimensional cyclic codes. Applications to decoding

We introduce the key equation of a multidimensional code. This equation exhibits the error-locator polynomial as product of univariate polynomials and the error-evaluator polynomial as a multivariate polynomial. Then we reinterpret these polynomials in a multidimensional linear recurring sequence context. In particular, using the concept of section, we reduce the solution of the decoding problem to a succession of application of the Berlekamp-Massey algorithm. However, it must be noted that multidimensional codes which are usefull for applications and which are decodable by our algorithm are left to be found

Genome wide association mapping of grain arsenic, copper, molybdenum and zinc in rice (Oryza sativa L.) grown at four international field sites

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