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A new lower bound on the family complexity of Legendre sequences

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Abstract

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.

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Acknowledgements

We would like to thank the anonymous reviewers for the detailed and carefully prepared suggestions, which improved not only the readability but also the quality of the paper. In particular, Corollary 2 is pointed out by the reviewers. The authors are supported by the Scientific and Technological Research Council of Turkey (TÜBÏTAK) under Project No: 116R026.

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  1. Hacettepe University, Department of Mathematics, Beytepe, 06800, Ankara, Turkey

    Yağmur Çakıroğlu & Oğuz Yayla

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  1. Yağmur Çakıroğlu
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  2. Oğuz Yayla
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Correspondence to Oğuz Yayla.

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Çakıroğlu, Y., Yayla, O. A new lower bound on the family complexity of Legendre sequences. AAECC 33, 173–192 (2022). https://doi.org/10.1007/s00200-020-00442-y

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