  Quantum contextuality takes an important place amongst the concepts of
quantum computing that bring an advantage over its classical counterpart. For a
large class of contextuality proofs, aka. observable-based proofs of the
Kochen-Specker Theorem, we formulate the contextuality property as the absence
of solutions to a linear system and define for a contextual configuration its
degree of contextuality. Then we explain why subgeometries of binary symplectic
polar spaces are candidates for contextuality proofs. We report the results of
a software that generates these subgeometries, decides their contextuality and
computes their contextuality degree for some small symplectic polar spaces. We
show that quadrics in the symplectic polar space $W_n$ are contextual for
$n=3,4,5$. The proofs we consider involve more contexts and observables than
the smallest known proofs. This intermediate size property of those proofs is
interesting for experimental tests, but could also be interesting in quantum
game theory.
