SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER 𝔽pm+u𝔽pm+u2𝔽pm

Title & Authors
SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER 𝔽pm+u𝔽pm+u2𝔽pm
Liu, Xiusheng; Xu, Xiaofang;

Abstract
Constacyclic codes of length $\small{p^s}$ over $\small{R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}}$ are precisely the ideals of the ring $\small{\frac{R[x]}{}$<$\small{x^{p^s}-1}$>$\small{}}$. In this paper, we investigate constacyclic codes of length $\small{p^s}$ over R. The units of the ring R are of the forms $\small{{\gamma}}$, $\small{{\alpha}+u{\beta}}$, $\small{{\alpha}+u{\beta}+u^2{\gamma}}$ and $\small{{\alpha}+u^2{\gamma}}$, where $\small{{\alpha}}$, $\small{{\beta}}$ and $\small{{\gamma}}$ are nonzero elements of $\small{\mathbb{F}_{p^m}}$. We obtain the structures and Hamming distances of all ($\small{{\alpha}+u{\beta}}$)-constacyclic codes and ($\small{{\alpha}+u{\beta}+u^2{\gamma}}$)-constacyclic codes of length $\small{p^s}$ over R. Furthermore, we classify all cyclic codes of length $\small{p^s}$ over R, and by using the ring isomorphism we characterize $\small{{\gamma}}$-constacyclic codes of length $\small{p^s}$ over R.
Keywords
constacyclic codes;cyclic codes;Hamming distance;repeated-root codes;
Language
English
Cited by
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