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 ${Ø}_K$, where $\ell$ is a square free integer such that $\ell \equiv 3\mathrm{mod}4$ and $C=[n,k]$ be a linear code defined over ${Ø}_K/2{Ø}_K$. The level $\ell$ theta function $\Th_{Ł_{\ell} (C)} $ of $C$ is defined on the lattice ${Ł}_{\ell }(C):=\{x\in {Ø}_K^n:{\rho }_{\ell }(x)\in C\}$, where ${\rho }_{\ell }:{Ø}_K\to {Ø}_K/2{Ø}_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell ,{\ell }^{\prime }$ such that $\ell \le {\ell }^{\prime }$, $\Th_{Λ_\ell}(q)$ and $\Th_{Λ_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac{\ell +1}{4}}$, % ii) for $\ell \ge \frac{2(n+1)(n+2)}{n}-1$, $\Th_{Ł_{\ell}} (C)$ 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 $\Th_{\La_\ell}(C)$.