Codes over $\mathbf {GF\pmb (4\pmb )}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ and Hermitian lattices over imaginary quadratic fields
Kok Seng Chua · Proceedings of the American Mathematical Society · 2004
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 $\textbf {GF(4)}$ 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 $\textbf {F}_2 \times \mathbf {F}_2$ 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 $\textbf {GF(4)}$ 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.