ON DESIGNS CONSTRUCTED FROM HADAMARD MATRICES SHOAIB UD DIN and V. C. MAVRON

There is a well-known correspondence between symmetric 2-(4ju— 1,2/z — 1,/z — 1) designs and Hadamard matrices of order 4^. It is not so well known that there is a natural correspondence between Hadamard matrices of order 2\x and affine l-(4ju, 2^, 2(i) designs whose duals are also affine. Such a design is denoted by H^fi) in this paper. Under this natural correspondence, two H^fi) designs are isomorphic if and only if their corresponding Hadamard matrices are equivalent. These H^fi) designs have been studied in various guises as, for example, symmetric nets, hypernets, symmetric resolvable transversal designs, orthogonal arrays, and elliptic semibiplanes. See [10,11,12,17]. More recently, under yet another name, Hadamard systems, they have been used to great effect to construct new families of symmetric 2-designs [18]. Using geometric-type arguments involving H^n) designs, we prove that if a Hadamard matrix of order n exists, then the number of equivalence classes of Hadamard matrices of order \i2 is at least n— 1. Hall [6] proved that there are exactly five equivalence classes of Hadamard matrices of order 16. Note that equivalent Hadamard matrices of the same order may correspond to non-isomorphic 2-(4^ — 1,2\i— l,ju— 1) designs. Using techniques developed in this paper, we show how five Hadamard matrices of order 16 can be obtained from a more general Hadamard matrix construction by combining two row or column equivalent copies of the Hadamard matrix of order 8. Then using Nandi's theorem we show how Hall's theorem follows from this. Essentially, we prove that any Hadamard matrix of order 16 must be of the form