Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let $K=Q(\sqrt{-\ell })$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$, and let $C=[n, k]$ is a linear code defined over $\mathcal{O}_K/2\mathcal{O}_K$. The level $\ell$ theta function $\Theta_{\Lambda_{\ell} (C) }$ of $C$ is defined on the lattice $\Lambda_{\ell} (C):= \{ x \in \mathcal{O}_K^n : \rho_\ell (x) \in C\}$, where $\rho_{\ell}:\mathcal{O}_K \rightarrow \mathcal{O}_K/2\mathcal{O}_K$ is the natural projection. In this paper, we prove that:

i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Theta_{\Lambda_\ell}(q)$ and $\Theta_{\Lambda_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$,

ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Theta_{\Lambda_{\ell}} (C)$ determines the code $C$ uniquely,

iii) for $\ell < \frac {2(n+1)(n+2)}{n} -1$, there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to $\Theta_{\Lambda_\ell}(C)$.