# New $2$-designs from strong difference families

## TL;DR

This paper advances the construction of 2-designs using strong difference families and cyclotomic conditions, leading to new existence results and improved bounds for specific combinatorial designs.

## Contribution

It introduces new methods to improve existence bounds for 2-designs derived from strong difference families, including seven newly established designs.

## Key findings

- Improved lower bounds on asymptotic existence of certain difference families.
- Extended Buratti's results for specific 2-designs with parameters involving 13 and 17.
- Established the existence of seven new 2-designs with specified parameters.

## Abstract

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of $(\mathbb{F}_{p}\times \mathbb{F}_{q},\mathbb{F}_{p}\times \{0\},k,\lambda)$-DFs for $k\in\{p,p+1\}$. We improve Buratti's existence results for $2$-$(13q,13,\lambda)$ designs and $2$-$(17q,17,\lambda)$ designs, and establish the existence of seven new $2$-$(v,k,\lambda)$ designs for $(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$, $(783,9,4)\}$.