This paper is concerned with a method based on birational geometry and produces dozens of new examples in codimensions 3, 4, 5 etc. The method is called unprojection by Reid. Using this method we construct new examples of K3 surfaces of codimensions 3 and 4 in weighted projective spaces from smaller codimension K3 surfaces whose rings are much simpler. This leads to the existence of almost all candidates for codimension 3 K3 surfaces in the list.
ALTINOK, SELMA (2005) "Constructing New K3 Surfaces," Turkish Journal of Mathematics: Vol. 29: No. 2, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol29/iss2/7