(\mathbb{Z}_p\mathbb{Z}_p[u]\)-additive codes

In this paper, we study \(\mathbb{Z}_p\mathbb{Z}_p[u]\)-additive codes, where \(p\) is prime and \(u^{2}=0\). In particular, we determine a Gray map from \( \mathbb{Z}_p\mathbb{Z}_p[u]\) to \(\mathbb{Z}_p^{ \alpha+2 \beta}\) and study generator and parity check matrices for these codes. We prove that a Gray map \(\Phi\) is a distance preserving map from (\(\mathbb{Z}_p\mathbb{Z}_p[u]\),Gray distance) to (\(\mathbb{Z}_p^{\alpha+2\beta}\),Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of \(\mathbb{Z}_p\mathbb{Z}_p[u]\)-additive cyclic codes.