We use the ring structure of $R$ to compute the remaining relations of $R$ as an $k$-algebra. This is done using the techniques introduced in [I. Stenger, A structure result for Gorenstein algebras of odd codimension. J. Algebra, 589:173–187, 2022]. The ideal $I_X$ defines the canonical model of a numerical Godeaux surface $X$ embedded in the weighted projective space $\mathbb{P}(2^2,3^4,4^4,5^3)$. The ring $R = S_{big}/I_X$ is the canonical ring of $X$.
Up to now, this procedure involves several smaller Hom-computations which are very time consuming.
The object canonicalRing is a method function.