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

Abstract

Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $\O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $\O_K/2\O_K$. The level $\ell$ theta function $\Th_{\L_{\ell} (C)} $ of $C$ is defined on the lattice $\L_{\ell} (C):= \set {x \in \O_K^n : \rho_\ell (x) \in C}$, where $\rho_{\ell}:\O_K \rightarrow \O_K/2\O_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Th_{\Lambda_\ell}(q)$ and $\Th_{\Lambda_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$, % ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Th_{\L_{\ell}} (C)$ 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 $\Th_{\La_\ell}(C)$.