Abstract An affine vector space partition of $${{\,\textrm{AG}\,}}(n,q)$$ AG ( n , q ) is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

Abstract For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of synchronising and separating are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of spreading permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an AB-Lemma) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

Abstract Let $${\mathcal {C}}$$ C be a 430-cap of $$\textrm{PG}(6,4)$$ PG ( 6 , 4 ) having two intersection sizes with respect to hyperplanes. We show that no hyperplane of $$\textrm{PG}(6,4)$$ PG ( 6 , 4 ) intersects $${\mathcal {C}}$$ C in a Hill 78-cap. So if it can be shown that the Hill 78-cap of $$\textrm{PG}(5,4)$$ PG ( 5 , 4 ) is projectively unique, then such a 430-cap does not exist, or equivalently, a two-weight $$[430,7]_{\mathbb {F}_4}$$ [ 430 , 7 ] F 4 linear code with dual weight at least 4, does not exist.