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 $\mathcal {O}_K$, 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 $\Theta _{\Lambda _\ell }(C)$.