W-complement d-spreads, singleton systems (Q1103866)

W\(\subseteq V\) are vector spaces over GF(q) of dimensions n and \(n+d\) respectively. A collection \(\Lambda\) of d-dimensional subspaces of V is defined to be a W-complement d-spread of V if for each \(A\in \Lambda\), \(A\cap W=\{0\}\) and \(0\neq v\in V\) such that \(<v>\cap W=\{0\}\) then \(<v>\) is contained in one and only one d-dimensional subspace lying in \(\Lambda\). The author establishes a one-to-one correspondence between the set of all W-complement d-spreads of V and the set of all (d,n,\(\ell,q)\)- singleton systems of order q n.