Partitioning the planes of AG2m(2) into 2-designs

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Abstract

A t-design Sλ(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 Sλ(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 Sλ(t′, k, v). It is shown that the S1(3, 4, 22m) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S1(2, 4, 22m) 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.

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The author was supported by a University Fellowship of the Ohio State University while conducting this research.

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Present Address: Department of Mathematics, University of Delaware, Newark, DE 19711, U.S.A.