\(\text{Double-}(2^n+1)\) configurations in \(\text{PG}(2n+1,2)\) (Q1266374)

This paper deals with the existence of \(\text{double-}(2^n+1)\) of \(n\)-spaces of some projective geometry \(P = \text{PG}(2n+1,2)\). This is a pair \((A,B)\) of sets \(A = \{A_0,\dots,A_{2^n}\}\) and \(B = \{B_0,\dots,B_{2^n}\}\) which consist of \(2^n+1\) pairwise disjoint subspaces of \(P\) of dimension \(n\) such that \(A_i \cap B_i\) has dimension \(n-1\) and \(A_i \cap B_j\) is a point for any \(i\neq j\), and \(\bigcup A = \bigcup B\). This definition is inspired by papers of \textit{R. Shaw} [ibid. 18, No. 3, 315-339 (1997; Zbl 0888.51008)], and R. Shaw and N. A. Gordon, who constructed a double-5 of planes in \(\text{PG}(5,2)\). The author shows that for any \(n\geq 2\) there exists a \(\text{double-}(2^n+1)\) of \(n\)-spaces in \(P\). Moreover, if \(n \geq 3\), there exist such configurations that are projectively inequivalent. This is done by determining the group of linear isomorphisms of these configurations. The key result is that the projective equivalence classes of \(\text{double-}(2^n+1)\) of \(n\)-spaces are in one-to-one correspondence with the \(\Aut(\text{GF} (2^n))\)-orbits in \(\text{GF} (2^n)\) of elements of trace 1.