  Maximum-distance separable (MDS) convolutional codes form an optimal family
of convolutional codes, the study of which is of great importance. There are
very few general algebraic constructions of MDS convolutional codes. In this
paper, we construct a large family of unit-memory MDS convolutional codes over
$\F$ with flexible parameters. Compared with previous works, the field size $q$
required to define these codes is much smaller. The construction also leads to
many new strongly-MDS convolutional codes, an important subclass of MDS
convolutional codes proposed and studied in \cite{GL2}. Many examples are
presented at the end of the paper.
