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

Abstract

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