\noindent The number of n-gaussoids is shown to be a double exponential function in n. The necessary bounds are achieved by studying construction methods for gaussoids that rely on prescribing 3-minors and encoding the resulting combinatorial constraints in a suitable transitive graph. Various special classes of gaussoids arise from restricting the allowed 3-minors.
In this paper, we explore a connection between binary hierarchical models, their marginal polytopes, and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.