Multi-qubit doilies: Enumeration for all ranks and classification for ranks four and five

For N?2, an N-qubit doily is a living in the symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending strategy Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving number both linear quadratic for any N>2. Then present effective algorithm generation all doilies. Using this N=4 N=5, provide classification terms types observables they feature negative lines endowed with. We also list distinguished findings about absent case, point out couple specific features exhibited by outline some prospective extensions our approach.