  It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are
two special classes of MDS codes with wide applications in practice. The
complete weight enumerators of these codes are very important for determining
the capability of both error-detection and error-correction. In this paper, for
any positive integer $m$ and prime $p$, basing on the character sums, we
determine the complete weight enumerators of the Reed-Solomon code and the
extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over
$\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding
results in \cite{BK91,K04}.
