  Both linear complementary dual (LCD) codes and maximum distance separable
(MDS) codes have good algebraic structures, and they have interesting practical
applications such as communication systems, data storage, quantum codes, and so
on. So far, most of LCD MDS codes have been constructed by employing
generalized Reed-Solomon codes. In this paper we construct some classes of new
Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially
equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed- Solomon
type. Our method is based on the constructions of Beelen et al. (2017) and Roth
and Lempel (1989). To the best of our knowledge, this is the first paper on the
construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of
non- Reed-Solomon type constructed by our method is not monomially equivalent
to any LCD code constructed by the method of Carlet et al. (2018).
