Codes over 𝐆𝐅\pmb(4\pmb) and 𝐅₂×𝐅₂ and Hermitian lattices over imaginary quadratic fields

Chua, Kok

doi:10.1090/S0002-9939-04-07724-X

Codes over $\mathbf {GF\pmb (4\pmb )}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ and Hermitian lattices over imaginary quadratic fields
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Proc. Amer. Math. Soc. 133 (2005), 661-670 Request permission

Abstract:

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.

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