Hierarchical Models, Marginal Polytopes, and Linear Codes
  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. 
Thomas Kahle, Walter Wenzel, Nihat Ay