  This work is devoted to the study Translation-Invariant (TI) Matrix Product
State (MPS) representations of quantum states with periodic boundary conditions
(PBC). We pursue two directions: we introduce new methods for constructing TI
MPS representations of a certain class of TI states and study their optimality
in terms of their bond dimension. We pay particular attention to the $n$-party
$W$-state and construct a TI MPS representation of bond dimension $\left
\lfloor \dfrac{n}{2} \right \rfloor +1$ for it. We further study properties of
this class and show that we can can always achieve a bond dimension of $n$ for
TI MPS representation of states in this class. In the framework of studying
optimality of TI MPS representations with PBC, we study the optimal bond
dimension $d(\psi)$ for a given state $\psi$. In particular we introduce a
deterministic algorithm for the search of $d(\psi)$ for an arbitary state.
Using numerical methods, we verify the optimality of our previous construction
for the $n$-party $W$-state for small $n$.
