Bounds on the number of Hadamard designs of even order (Q2748465)

Let \(V\) be a set of \(v\) elements (points) and let \(B\) be a collection of \(b\) \(k\)-subsets (blocks) of \(V\). The pair \((V,B)\) is called a 2-\((v,k,\lambda)\) design, if every pair of distinct points of \(V\) is contained in precisely \(\lambda\) blocks of \(B\). If \(b=v\), the design is called symmetric. A Hadamard design of order \(n\) is a symmetric 2-\((4n-1,2n-1,n-1)\) design. In this paper, the authors provide a new lower bound on the number of non-isomorphic Hadamard symmetric designs of even order. The new bound improves the bound on the number of Hadamard designs of order \(2n\) previously given by the authors by a factor of \(8n-1\) for every odd \(n > 1\), and for every even \(n\) such that \(4n-1 > 7\) is a prime. For orders 8, 10, and 12, the number of non-isomorphic Hadamard designs is shown to be at least 22,478,260, \(1.31 \times 10^{15}\), and \(10^{27}\), respectively. For orders \(2n=14,16,18\) and 20, a lower bound of \((4n-1)!\) is proved. It is conjectured that \((4n-1)!\) is a lower bound for all orders \(2n\geq 14\).