Bounds on the number of affine, symmetric, and Hadamard designs and matrices

Lower bouds on the number of non-isomorphic embeddings of a symmetric net i nto affine designs with classical parameters, of an affine design into symm etric 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 t he number of affine 2-(q(d), q(d-1),(q(d-1)-1)/(q - 1)) designs (d greater than or equal to3) that contain the classical (q, q(d-2))-net is improved b y a factor of q(3+4+...+d)(q - 1)(d-2) Similarly, the bound of Jungnickel f or the number of symmetric 2-((q(d+1) - 1)/(q - 1), (q(d) - 1)/(q - 1), (q( d-1) - 1)/(q - 1)) designs (d greater than or equal to 3) that contain the the classical affine design AG(d, q) as a residual design is improved to ma tch that of Kantor. Furthermore, for n large and by starting with rigid sym metric and affine designs, the lower bound for the number of non-isomorphic symmetric 2-(q(d+1) - 1)/(q - 1, (q(d) - 1)/(q - 1), (q(d-1) - 1)/(q - 1)) designs is improved to (q(d-1) + ... + q)!. By using the Paley design of o rder n = (q + 1)/4, q = 3 (mod 4) a prime power, a lower bound for the numb er of Hadamard designs of order q + 1 is also obtained. In particular, by c hoosing 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 impro ved from 389 to 1, 108, 800, and the bound on the number of affine 2-(64, 1 6, 5) designs is improved from 157 to 10, 810, 800. A similar method also i mproves the number of nonisomorphic Hadamard 2-(311 15, 7) designs from 1, 766, 891 to 11, 727, 788 and the number of non-isomorphic Hadamard 2-(39, 1 9, 9) designs from 38 to 5.87 x 10(14) The number of inequivalent Hadamard matrices of order 40 is at least 3.66 x 10(11) (C) 2000 Academic Press.