We consider the problem of representing quantum states as site independent matrix product states. We will show several examples and state the theorem that provides necessary and sufficient conditions for the representation of the W-state using matrix product states if one of the matrices has a particular form. Further, we will provide a pair of constructions: in addition to having closed boundary conditions, one of which has an improved proportionality constants for the W-state than ones known in the literature. We will also briefly touch upon the methods to prove the impossibility of representation of the W-state for specific number of qubits using matrices of fixed dimension using a classic theorem from algebraic geometry.