Some classes of repeated-root constacyclic codes over $mathbb F_{p^m}+umathbb F_{p^m}+u^2mathbb F_{p^m}$

摘要

Constacyclic codes of length $p^s$ over $R=FF_{p^m}+uFF_{p^m}+u^2FF_{p^m}$ are precisely the ideals of the ring $rac{R[x]}{langle x^{p^s}-1angle}$. In this paper, we investigate constacyclic codes of length $p^s$ over $R$. The units of the ring $R$ are of the forms $gamma$, $lpha+ueta$, $lpha+ueta+u^2gamma$ and $lpha+u^2gamma$, where $lpha$, $eta$ and $gamma$ are nonzero elements of $FF_{p^m}$. We obtain the structures and Hamming distances of all $(lpha+ueta)$-constacyclic codes and $(lpha+ueta+u^2gamma)$-constacyclic codes of length $p^s$ over $R$. Furthermore, we classify all cyclic codes of length $p^s$ over $R$, and by using the ring isomorphism we characterize $gamma$-constacyclic codes of length $p^s$ over $R$.