A Note on Hamming distance of constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$

Abstract

For any prime $p$, $\lambda$-constacyclic codes of length $p^s$ over $\mathcal{R}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ are precisely the ideals of the local ring ${\mathcal{R}}_{\lambda }=\frac{\mathcal{R}[x]}{⟨x^{p^s}-\lambda ⟩}$, where $u^2=0$. In this paper, we first investigate the Hamming distances of cyclic codes of length $p^s$ over $\mathcal{R}$. The minimum Hamming distances of all cyclic codes of length $p^s$ over $\mathcal{R}$ are determined. Moreover, an isometry between cyclic and $\alpha$-constacyclic codes of length $p^s$ over $\mathcal{R}$ is established, where $\alpha$ is a nonzero element of ${\mathbb{F}}_{p^m}$, which carries over the results regarding cyclic codes corresponding to $\alpha$-constacyclic codes of length $p^s$ over $\mathcal{R}$.