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Liu, Xiusheng,Xu, Xiaofang Korean Mathematical Society 2014 대한수학회지 Vol.51 No.4
Constacyclic codes of length $p^s$ over $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{<x^{p^s}-1>}$. 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 $\mathbb{F}_{p^m}$. 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.
SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER Fpm + uFpm + u2Fpm
Xiusheng Liu,Xiaofang Xu 대한수학회 2014 대한수학회지 Vol.51 No.4
Constacyclic codes of length ps over R = Fpm + uFpm + u2Fpm are precisely the ideals of the ring R[x]/(xps −1). In this paper, we investigate constacyclic codes of length ps over R. The units of the ring R are of the forms γ, α+uβ, α+uβ +u2γ and α+u2γ, where α, β and γ are nonzero elements of Fpm . We obtain the structures and Hamming distances of all (α+uβ)-constacyclic codes and (α+uβ+u2γ)-constacyclic codes of length ps over R. Furthermore, we classify all cyclic codes of length ps over R, and by using the ring isomorphism we characterize γ-constacyclic codes of length ps over R.