5,539 research outputs found

    Remarks on the k-error linear complexity of p(n)-periodic sequences

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    Recently the first author presented exact formulas for the number of 2ⁿn-periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k >2, of a random 2ⁿn-periodic binary sequence. A crucial role for the analysis played the Chan-Games algorithm. We use a more sophisticated generalization of the Chan-Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for pⁿn-periodic sequences over Fp, p prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of pⁿn-periodic sequences over Fp

    On the calculation of the linear complexity of periodic sequences

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    Based on a result of Hao Chen in 2006 we present a general procedure how to reduce the determination of the linear complexity of a sequence over a finite field \F_q of period unun to the determination of the linear complexities of uu sequences over \F_q of period nn. We apply this procedure to some classes of periodic sequences over a finite field \F_q obtaining efficient algorithms to determine the linear complexity

    Inconstancy of finite and infinite sequences

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    In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent than other criteria for measuring its variational complexity. We also compute the inconstancy of classical binary sequences including some automatic sequences and Sturmian sequences.Comment: Accepted by Theoretical Computer Scienc

    Discrete Dynamical Systems Embedded in Cantor Sets

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    While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with N N variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit NN\to\infty. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running top to bottom in figures, to appear in J. Math. Phy

    Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps

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    We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these dynamics defined by entropy conditions rather completely, obtaining a complete classification with respect to large entropy measures and a description of their measures with maximum entropy and periodic orbits. These results can in particular be applied to entropy-expanding maps like (x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the meromorphy of the Artin-Mazur zeta function on a large disk. This follows from a similar new result about strongly positively recurrent Markov shifts where the radius of meromorphy is lower bounded by an "entropy at infinity" of the graph.Comment: accepted by Annales de l'Institut Fourier, final revised versio
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