Nonexistence results for tight block designs

Dukes, Peter ; Short-Gershman, Jesse

Journal of Algebraic Combinatorics, Tome 38 (2013) no. 1, pp. 103-119.

Résumé

Summary: Recall that combinatorial 2s-designs admit a classical lower bound $b \ge\binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight 2s-designs for each fixed s$\geq 5$, although no concrete understanding of `finitely many' is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5\leq s\leq 9$, although the same analysis could in principle be extended to larger s. Additionally, we obtain strong necessary conditions for existence in the difficult case s=4.

Détail

Keywords: tight design, symmetric design, orthogonal polynomials, delsarte theory