When is there a unique socle-vector associated to a given $h$-vector?
  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. 
Fabrizio Zanello