On the Existence of q-Analogs of Steiner Systems
نویسندگان
چکیده
A q-analog of a Steiner system (briefly, q-Steiner system), denoted by S = Sq[t, k, n], is a set of k-dimensional subspaces of F n q such that each t-dimensional subspace of Fq is contained in exactly one element of S. Presently, q-Steiner systems are known only for t = 1 and in the trivial cases t = k and k = n. In this paper, the first known nontrivial q-Steiner systems with t ≥ 2 are constructed. Specifically, S2[2, 3, 13] q-Steiner systems are found by requiring that their automorphism group contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions.
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