Journal of the
Korean Mathematical Society JKMS

Constacyclic codes of length $p^s$ over $R=\FF_{p^m}+u\FF_{p^m}+u^2\FF_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{\langle x^{p^s}-1\rangle}$. In this paper, we investigate constacyclic codes of length $p^s$ over $R$. The units of the ring $R$ are of the forms $\gamma$, $\alpha+u\beta$, $\alpha+u\beta+u^2\gamma$ and $\alpha+u^2\gamma$, where $\alpha$, $\beta$ and $\gamma$ are nonzero elements of $\FF_{p^m}$. We obtain the structures and Hamming distances of all $(\alpha+u\beta)$-constacyclic codes and $(\alpha+u\beta+u^2\gamma)$-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 $R$.

Keywords: constacyclic codes, cyclic codes, Hamming distance, repeated-root codes

MSC numbers: 94B15, 94B05