Codes over rings of size $p^2$ and lattices over imaginary quadratic fields

Let $\mathrm{\ell }>0$ be a square-free integer congruent to 3 mod 4 and ${\text{O}}_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\mathrm{\ell }})$. Codes $C$ over rings ${\text{O}}_K\mathrm{/}p{\text{O}}_K$ determine lattices ${\mathrm{\Lambda }}_{\mathrm{\ell }}(C)$ over $K$. If $p\nmid \mathrm{\ell }$ then the ring $\mathbb{R}:={\text{O}}_K\mathrm{/}p{\text{O}}_K$ is isomorphic to ${\text{F}}_{p^2}$ or ${\text{F}}_p\times {\text{F}}_p$. Given a code $C$ over $\mathbb{R}$, theta functions on the corresponding lattices are defined. These theta series ${\theta }_{{\mathrm{\Lambda }}_{\mathrm{\ell }}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\mathrm{\ell }<{\mathrm{\ell }}^{\mathrm{\prime }}$ the first $\frac{\mathrm{\ell }+1}{4}$ terms of their corresponding theta functions are the same. Moreover, we conjecture that for $\mathrm{\ell }>\frac{p(n+1)(n+2)}{2}$ there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes $p<7$ and $\mathrm{\ell }\le 59$.