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

Efficient linear feedback shift registers with maximal period

We introduce and analyze an efficient family of linear feedback shift registers (LFSR's) with maximal period. This family is word-oriented and is suitable for implementation in software, thus provides a solution to a recent challenge posed in FSE '94. The classical theory of LFSR's is extended to provide efficient algorithms for generation of irreducible and primitive LFSR's of this new type

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-

On shortest linear recurrences

AbstractThis is an expository account of a constructive theorem on the shortest linear recurrences of a finite sequence over an arbitrary integral domainR. A generalization of rational approximation, which we call “realization”, plays a key role throughout the paper.We also give the associated “minimal realization” algorithm, which has a simple control structure and is division-free. It is easy to show that the number ofR-multiplications required isO(n2), wherenis the length of the input sequence.Our approach is algebraic and independent of any particular application. We view a linear recurring sequence as a torsion element in a naturalR[X]-module. The standardR[X]-module of Laurent polynomials overRunderlies our approach to finite sequences. The prerequisites are nominal and we use short Fibonacci sequences as running examples