Codes over GF(4) and F2 x F2 and Hermitian lattices over imaginary quadratic fields

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over GF(4) and F-2 x F-2, and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly l modular lattice Z + rootlZ. We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over F-2 x F-2 that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets GF(4) and F-2 x F-2 are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.