Abstract
A Singer cycle in GL(n,q) is an element of order q permuting cyclically all the nonzero vectors. Let σ be a Singer cycle in GL(2n,2). In this note we shall count the number of lines in PG (2n-1,2) whose orbit under the subgroup of index 3 in the Singer group 〈σ〉 is a spread. The lines constituting such a spread are permuted cyclically by the group 〈σ3〉, hence gives rise to a flag-transitive 2-(22n ,4,1) design.
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