Generalized parity proofs of the Kochen-Specker theorem

We discuss two approaches to producing generalized parity proofs of the Kochen-Specker theorem. Such proofs use contexts of observables whose product is $I$ or $-I$; we call them constraints. In the first approach, one starts with a fixed set of constraints and methods of linear algebra are used to produce subsets that are generalized parity proofs. Coding theory methods are used for enumeration of the proofs by size. In the second approach, one starts with the combinatorial structure of the set of constraints and one looks for ways to suitably populate this structure with observables. As well, we are able to show that many combinatorial structures can not produce parity proofs.