Repeated root cyclic codes over $\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$ and their Lee distances

Abstract

In this paper we have studied repeated root cyclic codes of length p^k over $R=\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$, u² = 0, where p is a prime and k is a positive integer. We have determined a unique set of generators for these codes and obtained some results on their Lee distances. A minimal spanning set for them has been obtained and their ranks are determined. Further, we have determined the complete algebraic structure of principally generated cyclic codes in this class. An upper bound for the Lee distance of linear codes over R is presented. We have considered two Gray maps $\psi :R \rightarrow \mathbb {Z}_{p}^{4}$ and $\phi _{1}:R \rightarrow \mathbb {Z}_{p^{2}}^{2}$, and using them, we have obtained some optimal binary linear codes as well as some quaternary linear codes from cyclic codes of length 4 over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$. Three of the quaternary linear codes obtained are new, and the remaining of them have the best known parameters for their lengths and types. We have also obtained some optimal ternary codes of length 12 as Gray images of repeated root cyclic codes of length 3 over $\mathbb {Z}_{9}+u\mathbb {Z}_{9}$.