The problem of obtaining an upper bound on the local reachability index I-LR for a positive 2-D system is still an unanswered question. Unfortunately, it is not clear what kind of mathematical instruments should be used to effectively tackle this problem. In this technical note, a tool is presented with the aim of facilitating the calculations of the reachability matrix and hence, the analysis of the local reachability property and of I-LR. Besides that, a new family of nth-order systems is constructed and, depending on the dimension n, an upper bound on I-LR of the corresponding systems is formulated.