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Multi-sequences with \(d\)-perfect property. (English) Zbl 1075.68024
Summary: Sequences with almost perfect linear complexity profile are defined by H. Niederreiter [in: Proceedings of the Salzburg Conference 1986, Teubner, Stuttgart, Contrib. Gen. Algebra 5, 221–233 (1987; Zbl 0641.65005)]. C. Xing and K. T. Lam [IEEE Trans. Inf. Theory 45, 1267–1270 (1999; Zbl 0943.94008)] and C. Xing [J. Complexity 16, 661–675 (2000; Zbl 1026.94006)] extended this concept from the case of single sequences to the case of multi-sequences and further proposed the concept of \(d\)-perfect multi-sequences. In this paper, based on the technique of \(m\)-continued fractions due to Dai et al., we investigate the property of \(d\)-perfect multi-sequences and obtain a sufficient and necessary condition for \(d\)-perfect multi-sequences. We show that \(d\)-perfect multi-sequences are not always strongly \(d\)-perfect. In particular, we give an example to disprove the conjecture, posed by Xing (2000), on \(d\)-perfect multi-sequences.

68P25 Data encryption (aspects in computer science)
Full Text: DOI
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[2] Z. Dai, K. Wang, D. Ye, Multidimensional Continued Fraction and Rational Approximation, http://arxiv.org/abs/math.NT/0401141.
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