On the Siamese Twin Designs

Abstract

Let 4n ² be the order of a Bush-type Hadamard matrix with q = (2n + 1)² a prime power. It is shown that there is a weighing matrix

$$ W(4)({q^{m}} + {q^{{m - 1}}} + \cdot \cdot \cdot + q + 1){n^{2}},4{q^{m}}{n^{2}}) $$

which can be used to construct a pair of symmetric designs with the parameters

$$ v = 4({q^{m}} + {q^{{m - 1}}} + \cdot \cdot \cdot + q + 1){n^{2}},{\text{ }}\kappa = {q^{m}}(2{n^{2}} + n),{\text{ }}\lambda = {q^{m}}({n^{2}} + n) $$

for every positive integer m. As a corollary we get a new class of symmetric designs with parameters

$$ v = 16({q^{m}} + {q^{{m - 1}}} + \cdot \cdot \cdot + q + 1){n^{2}},{\text{ }}\kappa {\text{ = }}{{\text{q}}^{{\text{m}}}}{\text{(8}}{{\text{n}}^{{\text{2}}}}{\text{ + 2n), }}\lambda {\text{ = }}{{\text{q}}^{{\text{m}}}}{\text{(4}}{{\text{n}}^{{\text{2}}}}{\text{ + 2n)}} $$

for all positive integers m and n, where 4n is the order a Hadamard matrix.

Thanks to W. Holzmann for his help and useful conversation.