Near vector spaces over GF(q) and (v,q+1,1)-BIBD's

Abstract The usual construction of ( v , q +1,1)−BIBD's from vector spaces over GF ( q ) is generalized to the class of near vector spaces over GF ( q ). It is shown that every ( v , q +1,1)−BIBD can be constructed from a near vector space over GF ( q ). Some corollaries are: Given a ( v 1 , q +1,1)−BIBD 〈 P 1 , B 1 〉 and a ( v 2 , q +1,1)−BIBD 〈 P 2 , B 2 〉, there is a (( q −1) v 1 v 2 + v 1 + v 2 , q +1,1)−BIBD 〈 P 3 , B 3 〉 containing 〈 P 1 , B 1 〉 and 〈 P 2 , B 2 〉 as disjoint subdesigns. If there is a ( v , q +1,1)−BIBD then there is a (( q −1) v +1, q ,1)−BIBD. Every finite partial ( v , q ,1)−BIBD can be embedded in a finite ( v ′, q +1,1)−BIBD.