Constructions of MDS Euclidean Self-dual Codes from smaller length

Systematic constructions of MDS Euclidean self-dual codes is widely concerned. In this paper, we consider the construction of MDS Euclidean self-dual codes from smaller length. We construct four new classes of $q$-ary MDS Euclidean self-dual codes from MDS Euclidean self-dual codes of known length via generalized Reed-Solomon (GRS for short) codes and extended GRS codes.