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 $\O_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $\O_K / p \O_K$ determine lattices $\Lambda_\ell (C) $ over $K$. If $ p \nmid \ell$ then the ring $\R:=\O_K / p \O_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, 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 $\ell \leq 59$.