Partitioning the planes of AG 2 m(2) into 2-designs

Abstract

A t-design S λ ( t, k, v) is an arrangement of v elements in blocks of k elements each such that every t element subset is contained in exactly λ blocks. A t-design S λ ( t, k, v) is called t′-resolvable if the blocks can be partitioned into families such that each family is the block system of a S λ ( t′, k, v). It is shown that the S 1(3, 4, 2 2 m ) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S 1(2, 4, 2 2 m ) given by the resolution is itself 1-resolvable. As a corollary it is shown that every odd dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation.