Some new large sets of geometric designs of type ${LS[3][2,3,2^8]}$
Keywords:
Geometric t-designs, Large sets of geometric t-designs, t-designs over GF(q), Parallelisms, Lattice basis reduction, LLL algorithmAbstract
Let $V$ be an $n$-dimensional vector space over $\F_q$. By a {\textit {geometric}} $t$-$[q^n,k,\lambda]$ design we mean a collection $\mathcal{D}$ of $k$-dimensional subspaces of $V$, called blocks, such that every $t$-dimensional subspace $T$ of $V$ appears in exactly $\lambda$ blocks in $\mathcal{D}.$ A {\it large set}, LS[N]$[t,k,q^n]$, of geometric designs, is a collection of N $t$-$[q^n,k,\lambda]$ designs which partitions the collection $V \brack k$ of all $k$-dimensional subspaces of $V$. Prior to recent article [4] only large sets of geometric 1-designs were known to exist. However in [4] M. Braun, A. Kohnert, P. \"{O}stergard, and A. Wasserman constructed the world's first large set of geometric 2-designs, namely an LS[3][2,3,$2^8$], invariant under a Singer subgroup in $GL_8(2)$. In this work we construct an additional 9 distinct, large sets LS[3][2,3,$2^8$], with the help of lattice basis-reduction.
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