We study h-sequences of certain Gorenstein standard G-algebras and, fo
r a given SI-sequence ($) under bar h = (h(1),..., h(s)) with h(1) = 3
, give a new construction of Gorenstein standard G-algebras with h-seq
uence ($) under bar h. Furthermore, using a similar idea, we give some
examples of Gorenstein SI-sequenceswith h(1) = 4.