Abstract
Combinatorial t-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a t-design. Till now only a small amount of work on constructing t-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of 2-designs, and explicitly determine their parameters. The obtained designs cover some earlier 2-designs as special cases. Furthermore, we confirm Conjecture 3 in Ding and Tang (ArXiv:1903.07375, 2019).
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Acknowledgements
The authors are very grateful to the reviewers and the Editor, for their comments and suggestions that improved the presentation and quality of this paper. The research of C. Xiang was supported by the National Natural Science Foundation of China (No. 11701187) and the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province of China (No. 2017A030310522). The research of X. Ling was supported by the National Natural Science Foundation of China (No. 11871058). The research of Q. Wang was supported by the National Natural Science Foundation of China under Grant No. 61672015.
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Xiang, C., Ling, X. & Wang, Q. Combinatorial t-designs from quadratic functions. Des. Codes Cryptogr. 88, 553–565 (2020). https://doi.org/10.1007/s10623-019-00696-9
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DOI: https://doi.org/10.1007/s10623-019-00696-9