Lattices in real quadratic fields and associated theta series arising from codes over ${\textbf{F}}_4$ and ${\textbf{F}}_2 \times {\textbf{F}}_2$

Abstract

Let $K = {\textbf{Q}}(\sqrt{d})$ with $d>0$ square-free and let ${\mathcal {O}}_{K}$ denote the ring of integers of K. Let ${\mathcal {C}} \subset {\mathcal {R}}^{n}$ be a linear code where ${\mathcal {R}}$ is ${\textbf{F}}_4$ if $d \equiv 5 \pmod {8}$ and ${\textbf{F}}_2 \times {\textbf{F}}_2$ if $d \equiv 1 \pmod {8}$. One has a surjective ring homomorphism $\rho : {\mathcal {O}}_{K}^{n} \rightarrow {\mathcal {R}}^n$ given by reduction modulo $(2{\mathcal {O}}_{K})^{n}$. The inverse image $\Lambda ({\mathcal {C}}):=\rho ^{-1}({\mathcal {C}})$ is a lattice associated to the code ${\mathcal {C}}$. One can associate to $\Lambda _{{\mathcal {C}}}$ a theta series $\Theta _{\Lambda _{d}({\mathcal {C}})}$. In this paper we consider how the theta series varies as one varies the value d. In particular, we show that for $d, d'$ with $d>d'$, one has $\Theta _{\Lambda _{d}({\mathcal {C}})} = \Theta _{\Lambda _{d^\prime }({\mathcal {C}})} + O\left( q^{\frac{d^\prime +1}{2}} \right) .$