We prove that the Nodal Severi variety parametrizing nodal quartic surfaces has a stratification comprising 34 irreducible components, whose incidence relation is described explicitly via a ‘genealogy tree’. Each component corresponds to a so-called shortening of the extended Kummer code. It is well-known that a quartic surface is a K3 surface: we then completely extend the above results to the Nodal Severi varieties parametrizing nodal K3 surfaces of all possible degrees. Moreover, we prove that the fundamental group of the smooth locus of a nodal K3 surface is determined by its associated code. We start by briefly reviewing the basics of coding theory, emphasizing the connections to nodal surfaces, and explaining how to associate a pair of binary codes \({\mathcal K}, {\mathcal K}^{\prime }\) to a nodal surface; main tools of this chapter are the Torelli theorem and Nikulin’s theory of primitive embeddings of lattices, while a key novel idea is to exploit the relationship between shortenings of codes and unobstructed deformations of nodal surfaces.