Codes over $\mathbf{GF\pmb(4\pmb)}$ and $\mathbf{F}_2 \times \mathbf{F}_2$ 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 ${\bf GF(4)}$ and $\mathbf{F}_2 \times \mathbf{F}_2$, and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly $\ell$ modular lattice $\mathbf{Z}+\sqrt{\ell}\mathbf{Z}$. We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over ${\bf F}_2 \times \mathbf{F}_2$ 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 ${\bf GF(4)}$ and $\mathbf{F}_2 \times \mathbf{F}_2$ are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.