Bounds on the Number of Affine, Symmetric, and Hadamard Designs and Matrices

AbstractLower bouds on the number of non-isomorphic embeddings of a symmetric net into affine designs with classical parameters, of an affine design into symmetric designs with classical parameters, and of a symmetric Hadamard design of order n into ones of order 2n are obtained. The bound of Jungnickel on the number of affine 2-(qd, qd−1, (qd−1−1)/(q−1)) designs (d⩾3) that contain the classical (q, qd−2)-net is improved by a factor of q3+4+…+d(q−1)d−2. Similarly, the bound of Jungnickel for the number of symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs (d⩾3) that contain the the classical affine design AG(d, q) as a residual design is improved to match that of Kantor. Furthermore, for d large and by starting with rigid symmetric and affine designs, the lower bound for the number of non-isomorphic symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs is improved to (qd−1+…+q)!. By using the Paley design of order n=(q+1)/4, q≡3 (mod4) a prime power, a lower bound for the number of Hadamard designs of order q+1 is also obtained. In particular, by choosing a non-classical net and non-classical affine design as the starting point, the bound on the number of symmetric 2-(40, 13, 4) designs is improved from 389 to 1, 108, 800, and the bound on the number of affine 2-(64, 16, 5) designs is improved from 157 to 10, 810, 800. A similar method also improves the number of non-isomorphic Hadamard 2-(31, 15, 7) designs from 1, 266, 891 to 11, 727, 788 and the number of non-isomorphic Hadamard 2-(39, 19, 9) designs from 38 to 5.87×1014. The number of inequivalent Hadamard matrices of order 40 is at least 3.66×1011