  We introduce self-dual codes over the Kleinian four group K = Z_2 x Z_2 for a
natural quadratic form on K^n and develop the theory.
  Topics studied are: weight enumerators, mass formulas, classification up to
length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs,
lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism
groups, marked codes.
  Kleinian codes form a new and natural fourth step in a series of analogies
between binary codes, lattices and vertex operator algebras. This analogy will
be emphasized and explained in detail.
