The resulting 'folding' produces a graph invariant for C via Pasch configurations and tensors. Moreover, the invariant is complete for Vasil'ev codes of length 15 as viewed by F. Hergert, showing the existence of nonadditive propelinear 1-perfect codes, and allowing to visualize a propelinear code by means of the commutative group formed by its classes mod kernel, as well as to generalize the notion of a propelinear code by extending the involved composition of permutations to a more general group product. Motivated by an application problem in computer architecture, Araujo, Dejter and Horak introduced a notion of perfect distance-dominating set, PDDS, in a graph, constituting a generalization of perfect Lee codes, diameter perfect codes, and other codes and dominating sets, and thus initiating a systematic study of such vertex sets. Some of these sets, related to the motivating application, were constructed, and the non-existence of others was demonstrated. In fact, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDSs, was stated. According to Dejter and Delgado, given a vertex subset S' of a side Pm of an m x n grid graph G, the perfect dominating sets S in G with S' being the intersection of S with V(Pm) can be determined via an exhaustive algorithm of running time O(2m+n). Extending the algorithm to infinite-grid graphs of width m-1, periodicity makes the binary decision tree prunable into a finite threaded tree, a closed walk of which yields all such sets S. The graphs induced by the complements of such sets S can be codified by arrays of ordered pairs of positive integers, for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes S (i.e. with just 1-cubes as induced components, also called  1-PDDS and DPL(2,4)), due to Klostermeyer and Goldwasser, allowed Dejter and Delgado to show that these sets S are restrictions of only one total perfect code S1 in the planar integer lattice graph, with the extra-bonus that the complement of S1 yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in the planar integer lattice graph are in 1-1 correspondence with the doubly infinite {0,1}-sequences.