  We attach two binary codes to a projective nodal surface (the strict code K
and, for even degree d, the extended code K' ) to investigate the `Nodal Severi
varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and
their incidence hierarchy, relating partial smoothings to code shortenings. Our
first main result solves a question which dates back over 100 years: the
irreducible components of F(4, n) are in bijection with the isomorphism classes
of their extended codes K', and these are exactly all the 34 possible
shortenings of the extended Kummer code K' , and a component is in the closure
of another if and only if the code of the latter is a shortening of the code of
the former. We extend this result classifying the irreducible components of all
nodal K3 surfaces in the same way, and we fully classify their extended codes.
In this classification there are some sporadic cases, obtain through projection
from a node.
  For surfaces of degree d=5 in P^3 we determine (with one possible exception)
all the possible codes K, and for several cases of K, we show the
irreducibility of the corresponding open set of F(5, n), for instance we show
the irreducibility of the family of Togliatti quintic surfaces. In the fourth
part we show that a `Togliatti-like' description holds for surfaces of degree 6
with the maximum number of nodes= 65: they are discriminants of cubic
hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an
irreducible 18-dimensional family of them. For degree d=6, our main result is
based on some novel auxiliary results: 1) the study of the half-even sets of
nodes on sextic surfaces, 2) the investigation of discriminants of cubic
hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K,
K' are uniquely determined, 4) the description of these codes, relating the
geometry of the Barth sextic with the Doro-Hall graph.
