Computations of linear rank inequalities on six variables

It is known that information inequalities on four random variables cannot be generated from a finite list. For the analogous case of linear rank variables, it is known that they can be generated from a finite list for up to five variables, but this is not known for six or more variables. Here we present partial results of computations on six-variable linear rank inequalities, showing that the number of sharp inequalities (those which cannot be generated from other inequalities) is more than one billion (counting variable-permuted forms). The problem is too large for standard polytope computation software; we describe the techniques used to generate and verify the current list of inequalities and a correspondingly large list of representable polymatroids. We also describe observed properties of the inequalities (some of which are now proven general results).