Large Sets of $t$-Designs over Finite Fields

Abstract

A $t\text{-}(n,k,\lambda ;q)$-design is a set of $k$-subspaces, called blocks, of an $n$-dimensional vector space $V$ over the finite field with $q$ elements such that each $t$-subspace is contained in exactly $\lambda$ blocks. A partition of the complete set of $k$-subspaces of $V$ into disjoint $t\text{-}(n,k,\lambda ;q)$ designs is called a large set of $t$-designs over finite fields. In this paper we give the first nontrivial construction of such a large set with $t\ge 2$.

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