Multi-sequences with \(d\)-perfect property.

*(English)*Zbl 1075.68024Summary: 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.

##### MSC:

68P25 | Data encryption (aspects in computer science) |

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\textit{X. Feng} et al., J. Complexity 21, No. 2, 230--242 (2005; Zbl 1075.68024)

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##### References:

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[9] | Xing, C., Multi-sequences with almost perfect linear complexity profile and function fields over finite fields, J. complexity, 16, 661-675, (2000) · Zbl 1026.94006 |

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