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

TL;DR

This paper explores the relationship between codes over rings of size p^2 and lattices over imaginary quadratic fields, analyzing their theta functions and conjecturing uniqueness conditions for their complete weight enumerators.

Contribution

It establishes a connection between codes over rings and lattices, expresses theta functions via weight enumerators, and proposes a conjecture on uniqueness for large parameters.

Findings

Theta functions are expressed in terms of complete weight enumerators.

For different ll, the initial terms of theta functions are identical.

Conjecture on the uniqueness of weight enumerators for large ll, verified for small primes.

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 $\ell \leq 59$.