In lattice-based cryptographic schemes, both encoded messages and accumulated decryption noise are represented in a modulo $q$ space. Therefore, it is natural to study toroidal distances and maximum toroidal distance (MTD) codes. In this paper, we derive some upper bounds for minimum toroidal distance of a code, including a Plotkin-type bound, a local ball--Plotkin bound, and a Delsarte linear programming bound. We also exhibit examples showing that these bounds are sharp in some cases. Moreover, we present several code constructions with good minimum distance, some of which are MTD codes. For $\ell=2$, we obtain a family of four-point MTD codes in $\mathbb Z_q^2$. For $\ell=4$, we propose a general code construction and exhibit several explicit instances for specific values of $q$, some of which are proven to be MTD codes. For $\ell=8$, using the $E_8$ lattice, we construct codes $C=2mE_8\cap \mathbb Z_q^8$, where $q=4m$ and show that they are MTD codes. These results give explicit optimal constructions of MTD codes for $\ell=2,4,8$. In the case $\ell=16$, we construct a code with minimum toroidal distance $3$ for $q=4$, while the known upper bound in this case is $2\sqrt{3}$. Our main tools are geometric and linear programming methods.

翻译：在基于格的密码方案中，编码消息和累积解密噪声均表示在模$q$空间中，因此研究环面距离和最大环面距离码具有重要意义。本文推导了码的最小环面距离的一些上界，包括Plotkin型界、局部球-Plotkin界和Delsarte线性规划界。我们还给出了示例，表明这些界在某些情况下是紧的。此外，我们提出了几种具有良好最小距离的码构造，其中一些是最大环面距离码。对于$\ell=2$，我们在$\mathbb Z_q^2$中得到了一个四点最大环面距离码族。对于$\ell=4$，我们提出了一般的码构造，并给出了$q$取特定值时的几个显式实例，其中一些被证明是最大环面距离码。对于$\ell=8$，利用$E_8$格，我们构造了码$C=2mE_8\cap \mathbb Z_q^8$（其中$q=4m$），并证明它们是最大环面距离码。这些结果给出了$\ell=2,4,8$情形下显式的最优最大环面距离码构造。对于$\ell=16$，我们构造了当$q=4$时最小环面距离为$3$的码，而已知上界为$2\sqrt{3}$。我们的主要工具是几何方法和线性规划方法。