A place cell is a neuron corresponding to a subset of Euclidean space known as a place field, that will fire if and only if the individual possessing the place cell is within that place field. The firing patterns of a collection of n place cells can be represented by a neural code C on n neurons, which is a subset of the power set of {1,2,...,n}. Determining whether C is convex, meaning that there is an arrangement of convex open place fields for which C is the code, remains an open problem. A sufficient condition for convexity is being max intersection complete: any intersection of maximal elements of C is also an element of C. Currently, the only way to determine this property is to evaluate all such intersections. We present a new method to determine max intersection completeness by introducing a simplicial complex for C called the factor complex of C. We show how to construct the factor complex using Stanley-Reisner theory, describe how this complex encodes information about C, and give an algorithm to check whether C is max intersection complete using the factor complex of a closely related code.