5-designs in affine spaces

Abstract

The ^-dimensional affine group over GF{2) is triply transitive on 2 n symbols. For n ^ 4, 4 ^ k ^ 2 n ~\ any orbit of ^-subsets is a 3-(2 , k, ) design. In this paper a sufficient condition that such a design be a 4-design is given. It is also shown that such a 4-design must always be a 5-design. A 5-design on 256 varieties with block size 24 is constructed in this fashion.