q-Analogs of Steiner Systems

Abstract

A Steiner structure $\dS = \dS_q[t,k,n]$ is a set of $k$-dimensional subspaces of $\F_q^n$ such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one subspace of $\dS$. Steiner structures are the $q$-analogs of Steiner systems; they are presently known to exist only for $t = 1$, $t=k$, and\linebreak for $k = n$. The existence of nontrivial $q$-analogs of Steiner systems has occupied mathematicians for over three decades. In fact, it was conjectured that they do not exist. In this paper, we show that nontrivial Steiner structures do exist. First, we describe a general method which may be used to produce Steiner structures. The method uses two mappings in a finite field: the Frobenius map and the cyclic shift map. These maps are applied to codes in the Grassmannian, in order to form an automorphism group of the Steiner structure. Using this method, assisted by an exact-cover computer search, we explicitly generate a Steiner structure $\dS_2[2,3,13]$. We conjecture that many other Steiner structures, with different parameters, exist.