An enumeration of certain projective ternary two-weight codes and their relationship to the cubic Segre variety

We detail the enumeration of all two-intersection sets of the five-dimensional projective space over the field of order 3 that are invariant under an element of order 7, which include the examples of Hill (1973) and Gulliver (1996). Up to projective equivalence, there are 6635 such two-intersection sets.