Designs over finite fields by difference methods

TL;DR

This paper uses difference methods to prove the existence of cyclic designs over finite fields, extending previous results to all odd integers and discovering new infinite families of group divisible designs over .

Contribution

It generalizes earlier results by showing cyclic 2-$(n,3,7)$ designs exist for all odd n and introduces the first infinite family of such designs over .

Findings

Proves cyclic 2-$(n,3,7)$ designs exist for all odd n.

Identifies the first infinite family of non-trivial cyclic group divisible designs over .

Enhances understanding of design existence over finite fields.

Abstract

One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and improve a little bit this result showing that it is true, more generally, for every odd $n$. In this way, we also find the first infinite family of non-trivial cyclic group divisible designs over $\mathbb{F}_{2}$.