Codes over rings of size four, Hermitian lattices, and corresponding theta functions

TL;DR

This paper explores the relationship between theta functions of lattices derived from codes over rings of size four, establishing conditions under which these functions determine the codes uniquely or form families of equivalent polynomials.

Contribution

It proves new bounds linking theta functions to code determination and describes the behavior of these functions across different levels for codes over rings of size four.

Findings

Theta functions have identical coefficients up to a certain power for levels with ll  

High-level theta functions uniquely determine the code

Low-level theta functions correspond to a family of symmetrized weight enumerator polynomials

Abstract

Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $\O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $\O_K/2\O_K$. The level $\ell$ theta function $\Th_{\L_{\ell} (C)} $ of $C$ is defined on the lattice $\L_{\ell} (C):= \set {x \in \O_K^n : \rho_\ell (x) \in C}$, where $\rho_{\ell}:\O_K \rightarrow \O_K/2\O_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Th_{\Lambda_\ell}(q)$ and $\Th_{\Lambda_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$, % ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Th_{\L_{\ell}} (C)$ 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 $\Th_{\La_\ell}(C)$.