On affine designs and Hadamard designs with line spreads Original Research Article

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291–303] described a construction that relates any Hadamard design H on image points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in image. Here it is proved that the affine design is the classical design of points and hyperplanes in image if, and only if, H is the classical design of points and hyperplanes in image and the line spread is of a special type. Computational results about line spreads in image are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in image, and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153–226].