The code problem for traces — improving the boundaries

The code problem for traces - given a finite set X, decide whether every element in X(+) has a unique factorization over X - is decidable if the independence relation equals P-4, the line graph on four nodes. Additionally, it is undecidable for a particular independence relation that does not have C-4, the cycle on four nodes, as induced subgraph. These results improve on the previously known boundaries of(un)decidable cases for this problem.