STS-graphs of perfect codes mod kernel

We show that a perfect 1-error-correcting code C is 'foldable' over its kernel via the Steiner triple systems associated to its codewords. 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, showing among other things 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.