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|q-1/2, n|q+1/2 and a family of negacyclic LCD codes of length n = q - 1. Furthermore, we obtain five families of q2-ary Hermitian MDS negacyclic LCD codes of length n|(q-1) and four families of Hermitian negacyclic LCD codes of length n = q2 + 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 q2-cyclotomic classes we give lower bounds on the minimum distance.

In this work, we study cyclic codes that have generators as Fibonacci polynomials over finite fields. We show that these cyclic codes in most cases produce families of maximum distance separable and optimal codes with interesting properties. We explore these relations and present some examples. Also, we present applications of these codes to secret sharing schemes.