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تسجيل مستخدم جديدA new series of large sets of subspace designs over the binary field
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In this article, we show the existence of large sets $operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v geq 8$ and $0 leq k leq v$ such that for the remainders $bar{v}$ and $bar{k}$ of $v$ and $k$ modulo $6$ we have $2 leq bar{v} < bar{k} leq 5$. The proof is constructive and consists of two parts. First, we give a computer construction for an $operatorname{LS}_2[3](2,4,8)$, which is a partition of the set of all $4$-dimensional subspaces of an $8$-dimensional vector space over the binary field into three disjoint $2$-$(8, 4, 217)_2$ subspace designs. Together with the already known $operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Gra{ss}mannian into joins yields a construction for the claimed large sets.
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In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace d
esigns. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $operatorname{LS}_3[2](2,k,v)$ and $operatorname{LS}_5[2](2,k,v)$ with integers $vgeq 6$, $vequiv 2mod 4$ and $3leq kleq v-3$, $kequiv 3mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.
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A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter
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