Nontrivial t-designs over finite fields exist for all t

摘要

A t-(n, k, A) design over IF, is a collection of k-dimensional subspaces of F-q(n), called blocks, such that each t-dimensional subspace of F-q(n). is contained in exactly lambda blocks. Such t-designs over F-q are the q-analogs of conventional combinatorial designs. Nontrivial t-(n, k, lambda) designs over F-q are currently known to exist only for t <= 3. Herein, we prove that simple (meaning, without repeated blocks) nontrivial t-(n, k, lambda) designs over F-q exist for all t and q, provided that k > 12(t+1) and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs. (C) 2014 Elsevier Inc. All rights reserved.