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