On affine designs and Hadamard designs with line spreads

Rahilly [10] described a construction that relates any Hadamard design H on 4m−1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m, 4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m, 4) if, and only if, H is the classical design of points and hyperplanes in PG(2m−1, 2) and the line spread is of a special type. Computational results about line spreads in PG(5, 2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3, 4), and provides a counter-example to a conjecture of Hamada [6]. Dedicated to Jennifer Seberry on her 60th birthday