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Systematic construction of \(q\)-analogs of \(t\)-\((v,k,\lambda)\)-designs. (English) Zbl 1055.05009
Summary: We consider a \(q\)-analog of \(t\)-\((v,k,\lambda)\)-designs. It is canonic since it arises by replacing sets by vector spaces over GF\((q)\), and their orders by dimensions. These generalizations were introduced by S. Thomas [Geom. Dedicata 63, No. 3, 247–253 (1996; Zbl 0863.51011)], they are called \(t\)-\((v,k,\lambda;q)\)-designs. A few of such \(q\)-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package.
In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on \(t\)-\((v,k,\lambda)\)-designs on sets, obtaining this way new results on the existence of \(t\)-\((v, k,\lambda;q)\)-designs on spaces for further quintuples \((t,v,k,\lambda;q)\) of parameters. We present several 2-\((6,3,\lambda;2)\)-designs, 2-\((7,3,\lambda;2)\)-designs and, as far as we know, the very first 3-designs over GF\((q)\).

05B05 Combinatorial aspects of block designs
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