On recursive constructions for 2-designs over finite fields

Abstract

This paper concentrates on recursive constructions for 2-designs over finite fields. In 1998, Itoh presented a powerful recursive construction: for certain index λ, if there exists a Singer cycle invariant 2-${(l,3,\lambda )}_q$ design, then there also exists an SL$(m,q^l)$ invariant 2-${(ml,3,\lambda )}_q$ design for all integers $m\ge 3$. We investigate the $\mathrm{GL}(m,q^l)$-incidence matrix between 2-subspaces and k-subspaces of $\mathrm{GF}{\left(q\right)}^{ml}$ with $m\ge 2$ and $k\ge 3$ in this work. As a generalization of Itoh's construction, the important case of $m=2$ is supplemented and a doubling construction is established for 2-${(l,3,\lambda )}_q$ designs over finite fields. As a further generalization, a product construction is developed for q-analogs of group divisible designs (q-GDDs). For general block dimension $k\ge 3$, several new infinite families of q-GDDs are constructed. As applications, plenty of new infinite families of 2-designs over finite fields are constructed.