Let q be a prime power and ${\mathbb{F}}_q$ the finite field with q elements. In this paper, it is shown that the minimal polynomial of a modified de Bruijn sequence of order n over ${\mathbb{F}}_q$ cannot be the product of an irreducible polynomial of degree n over ${\mathbb{F}}_q$ and any polynomial of degree k over ${\mathbb{F}}_q$ if $n\ge 4k$, which in fact improves several previous results ([5], [13]). Based on this result, a non-trivial lower bound $5n/4$ is given for linear complexities of modified de Bruijn sequences of order n distinct from m-sequences, which is the first non-trivial lower bound with no restriction on n.

中文翻译：

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重新讨论了修饰的de Bruijn序列的最小多项式

令q为素数${\mathbb{F}}_q$具有q个元素的有限域。在本文中，它表明的变形日的顺序Bruijn的序列中的最小多项式Ñ过${\mathbb{F}}_q$不能是n阶以上的不可约多项式的乘积${\mathbb{F}}_q$以及k之上的任何多项式${\mathbb{F}}_q$ 如果 $ñ\ge 4ķ$，实际上改善了先前的几个结果（[5]，[13]）。根据此结果，得出一个非平凡的下界$5ñ/4$给出了与m序列不同的n阶修饰de Bruijn序列的线性复杂度，m序列是对n无限制的第一个非平凡下界。