A code $C$ is called $\Z_p\Z_{p^2}$-linear if it is the Gray image of a $\Z_p\Z_{p^2}$-additive code, where $p>2$ is prime. In this paper, the rank and the dimension of the kernel of $\Z_p\Z_{p^2}$-linear codes are studied. Two bounds of the rank of a $\Z_3\Z_{9}$-linear code and the dimension of the kernel of a $\Z_p\Z_{p^2}$-linear code are given, respectively. For each value of these bounds, we give detailed construction of the corresponding code. Finally, pairs of rank and the dimension of the kernel of $\Z_3\Z_{9}$-linear codes are also considered.

翻译：如果代码是美元=p ⁇ p ⁇ p ⁇ 2}美元额外代码的灰色图像,则称之为美元=线性代码。在本文件中,研究了美元=p ⁇ p ⁇ 2}美元线性代码的等级和层面。如果代码是美元=p ⁇ p ⁇ p ⁇ 2}美元额外代码的灰色图像,则称之为美元-线性代码。我们分别给出了美元-线性代码的大小。我们对这些界限的每个值进行详细的构建。最后,对等代码的等级和层面也考虑了美元=3 ⁇ 9}线性代码的大小。