In this paper we study a family of Legendre sequences and its pseudo-randomness in terms of their family complexity. We present an improved lower bound on the family complexity of a family based on the Legendre symbol of polynomials over a finite field. The new bound depends on the Lambert W function and the number of elements in a finite field belonging to its proper subfield. Moreover, we present another lower bound which is a simplified version and approximates the new bound. We show that both bounds are better than previously known ones.

中文翻译：

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勒让德序列家族复杂度的新下界

在本文中，我们研究了勒让德序列族及其族复杂性的伪随机性。基于有限域上多项式的勒让德符号，我们提出了一个改进的家族复杂度下界。新边界取决于 Lambert W 函数和属于其适当子域的有限域中的元素数。此外，我们提出了另一个下界，它是一个简化版本并近似于新界。我们表明这两个边界都比以前已知的更好。