We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is equal or similar to $J_n$, the Jordan block with eigenvalue 1 and dimension n. If $M=J_2$, we classify all digit sets of size two allowing representation for all of ${\mathbb{Z}}^2$. For $M=J_n$ with $n\ge 3$, we show that a digit set of size three suffice to represent all of ${\mathbb{Z}}^n$. For bases M similar to $J_n$, $n\ge 2$, we construct a digit set of size n such that all of ${\mathbb{Z}}^n$ is represented. The language of words representing the zero vector with $M=J_2$ and the digits ${(0,\pm 1)}^T$ is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

我们研究具有矩阵基M和向量数字的数字系统中积分向量的表示。我们关注M等于或相似的情况$J_n$，特征值为 1 且维度为n的 Jordan 块。如果$\mathrm{中号}=J_2$，我们对所有大小为 2 的数字集进行分类，允许表示所有${\mathbb{Z}}^2$。为了$\mathrm{中号}=J_n$和$n\ge 3$，我们证明大小为 3 的数字集足以表示所有${\mathbb{Z}}^n$。对于碱基M类似于$J_n$,$n\ge 2$，我们构造一个大小为n的数字集，使得所有${\mathbb{Z}}^n$被代表。表示零向量的单词的语言$\mathrm{中号}=J_2$和数字${（0,\pm 1）}^{\mathrm{时间}}$被证明不是上下文无关的，而是可以被具有对数内存的图灵机识别。