  We perform an exact enumeration study of polymers formed from a (quenched)
random sequence of charged monomers $\pm q_0$, restricted to a 2-dimensional
square lattice. Monomers interact via a logarithmic (Coulomb) interaction. We
study the ground state properties of the polymers as a function of their excess
charge $Q$ for all possible charge sequences up to a polymer length N=18. We
find that the ground state of the neutral ensemble is compact and its energy
extensive and self-averaging. The addition of small excess charge causes an
expansion of the ground state with the monomer density depending only on $Q$.
In an annealed ensemble the ground state is fully stretched for any excess
charge $Q>0$.
