  Linear codes with complementary duals (LCD) have a great deal of significance
amongst linear codes. Maximum distance separable (MDS) codes are also an
important class of linear codes since they achieve the greatest error
correcting and detecting capabilities for fixed length and dimension. The
construction of linear codes that are both LCD and MDS is a hard task in coding
theory. In this paper, we study the constructions of LCD codes that are MDS
from negacyclic codes over finite fields of odd prime power $q$ elements. We
construct four families of MDS negacyclic LCD codes of length
$n|\frac{{q-1}}{2}$, $n|\frac{{q+1}}{2}$ and a family of negacyclic LCD codes
of length $n=q-1$. Furthermore, we obtain five families of $q^{2}$-ary
Hermitian MDS negacyclic LCD codes of length $n|\left( q-1\right)$ and four
families of Hermitian negacyclic LCD codes of length $n=q^{2}+1.$ For both
Euclidean and Hermitian cases the dimensions of these codes are determined and
for some classes the minimum distances are settled. For the other cases, by
studying $q$ and $q^{2}$-cyclotomic classes we give lower bounds on the minimum
distance.
