Derived and residual subspace designs. (English) Zbl 1314.51002
Summary: A generalization of forming derived and residual designs from \(t\)-designs to subspace designs is proposed. A \(q\)-analog of a theorem by Tran Van Trung [Arch. Math. 47, 187–192 (1986; Zbl 0608.05010)], D. C. van Leijenhorst [J. Comb. Theory, Ser. A 31, 146–154 (1981; Zbl 0481.05020)] and L. M. H. E. Driessen [\(t\)-designs, \(t\geq 3\). Techn. Rep., Technische Universiteit Eindhoven (1978)] is proven, stating that if for some (not necessarily realizable) parameter set the derived and residual parameter set are realizable, the same is true for the reduced parameter set.
As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no \(q\)-analog of the large Witt design.
As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no \(q\)-analog of the large Witt design.
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 05B05 | Combinatorial aspects of block designs |
| 05B25 | Combinatorial aspects of finite geometries |
Keywords:
\(q\)-analog; combinatorial design; subspace design; derived design; residual design; large setReferences:
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