Bounds on the number of affine, symmetric, and Hadamard designs and matrices
Journal of Combinatorial Theory Series A
Polarities, quasi-symmetric designs, and Hamada's conjecture
Designs, Codes and Cryptography
Designs having the parameters of projective and affine spaces
Designs, Codes and Cryptography
On the number of designs with affine parameters
Designs, Codes and Cryptography
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The hyperplanes in the affine geometry AG(d, q) yield an affineresolvable design with parameters 2-(q^d, q^{d-1}, \frac{q^{d-1} -1}{q-1}). Jungnickel [3]proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parametersfor d ≥ 3. The bound of Jungnickel was improved recently [5] by a factor ofq^{\frac{d^2+d-6}{2}}(q-1)^{d-2} for any d ≥ 4. In this paper, a construction of2-(q^d, q^{d-1}, \frac{q^{d-1}-1}{q-1}) designs based on group divisible designs is given that yieldsat least \frac{(q^{d-1}+q^{d-2}+\cdots+1)!(q-1)}{|{\rm P}\Gamma {\rm L}(d,q)||{\rm A}\Gamma{\rm L}(d,q)|} non-isomorphic designs for any d ≥ 3. This new bound improves the bound of[5] by a factor of \frac{1}{q^d}\prod_{i=1}^{(q^{d-1}-q)/(q-1)}(q^{d-1}+i).For any given q and d, It was previously known [2,11] that there are at least 8071non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is atleast 245,100,000.