Designs over finite fields by difference methods

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 ${\mathbb{F}}_2$.