A code $C$ is called $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear if it is the Gray image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For any prime number $p$ larger than $3$, the bounds of the rank of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear codes are given. For each value of the rank and the pairs of rank and the dimension of the kernel of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear codes, we give detailed construction of the corresponding codes. Finally, as an example, the rank and the dimension of the kernel of $\mathbb{Z}_5\mathbb{Z}_{25}$-linear codes are studied.
翻译：如果代码是$\mathb ⁇ p\mathb ⁇ p\\mathb ⁇ p}$-addive 代码的灰色图像,则代号为$C$( mathb ⁇ p\mathb}p\mathb ⁇ p}p ⁇ 2}$线性代码。对于任何数额大于$mathb ⁇ p}p\mathb ⁇ p\\mathb ⁇ p}p ⁇ 2}的质数,则代号为$\mathb ⁇ p\mathb ⁇ p\mathb ⁇ p}$-linear 代码的灰色图像,则代号为$\mathbb}2}(美元)内核代码的大小,则称为$\mathbb ⁇ 5\mathb}25$-线性代码的等级和维度,我们给出了相应的代码的详细构建。最后,举例来说,研究的是$\mathbb ⁇ 5\\mathb}25美元内核代码的等级和维度。
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