# Combinatorial t-designs from quadratic functions

## TL;DR

This paper explores the construction of infinite families of 2-designs from quadratic functions over finite fields, explicitly determining their parameters and confirming a recent conjecture, thus advancing combinatorial design theory.

## Contribution

It introduces a new method for constructing infinite families of 2-designs using quadratic functions and explicitly determines their parameters, confirming a recent conjecture.

## Key findings

- Constructed infinite families of 2-designs from quadratic functions.
- Explicitly determined parameters of the obtained designs.
- Confirmed Conjecture 3 in Ding and Tang (2019).

## 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 confirmed Conjecture $3$ in Ding and Tang (arXiv: 1903.07375, 2019).