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 confirmed Conjecture $3$ in Ding and Tang (arXiv: 1903.07375, 2019).
翻译:几十年来,组合型$t-设计一直是组合型设计中一个有趣的主题,最近有报道称,某些特殊多元型设计固定规模的图像组可能构成美元设计。直到现在,只完成了少量从特殊多元型设计中建造美元设计的工作,一般很难确定参数。在本文中,我们进一步调查这一想法,利用对有限字段的二次函数,从而获得无限的2美元设计组,并明确确定其参数。获得的设计覆盖了早先的2美元设计作为特例。此外,我们还确认在丁和唐的预测值为3美元(arXiv:1903.07375,2019美元)。