In the paper, we firstly study the algebraic structures of $\mathbb{Z}_p \mathbb{Z}_{p^k}$-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for a class of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes whose Gray images are linear (not necessarily cyclic) over $\mathbb{Z}_p$ is given. Moreover, as for some special families of cyclic codes over $\mathbb{Z}_{9}$ and $\mathbb{Z}_3 \mathbb{Z}_{9}$, the linearity of the Gray images is determined.
翻译：在论文中,我们首先研究了 $mathbb ⁇ p \ mathb ⁇ p ⁇ k}$-additive cyclic code 的代数结构, 并给出了生成多元代码和这些代码的最小范围 。 其次, 对于某类 $mathb ⁇ p\ mathb ⁇ p\ mathb ⁇ p ⁇ 2}$- addific code, 其灰色图像为线性( 不一定周期性) 超过$\ mathbb ⁇ p ⁇ p 。 此外, 对于 超过$\ mathb ⁇ 9} 和$\ mathbb} 3\ mathb ⁇ 9} 的周期代码的某些特殊家庭, 灰色图像的线性已经确定 。
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