  First, we construct a bijection between the set of $h$-vectors and the set of
socle-vectors of artinian algebras. As a corollary, we find the minimum
codimension that an artinian algebra with a given socle-vector can have. Then,
we study the main problem in the paper: determining when there is a unique
socle-vector for a given $h$-vector. We solve the problem completely if the
codimension is at most 3.
