  MDS self-dual codes over finite fields have attracted a lot of attention in
recent years by their theoretical interests in coding theory and applications
in cryptography and combinatorics. In this paper we present a series of MDS
self-dual codes with new length by using generalized Reed-Solomon codes and
extended generalized Reed-Solomon codes as the candidates of MDS codes and
taking their evaluation sets as an union of cyclotomic classes. The conditions
on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.
