  Systematic constructions of MDS self-dual codes is widely concerned. In this
paper, we consider the constructions of MDS Euclidean self-dual codes from
short length. Indeed, the exact constructions of MDS Euclidean self-dual codes
from short length ($n=3,4,5,6$) are given. In general, we construct more new of
$q$-ary MDS Euclidean self-dual codes from MDS self-dual codes of known length
via generalized Reed-Solomon (GRS for short) codes and extended GRS codes.
