Applications of Coding Theory to the Construction of Modular Lattices

We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the codes reaching this bound, and derive some extremal lattices of levell=2, 3, 7 and minimum 4, 6, 8.