A reduced-order observer is designed for a class of linear discrete-time Markovian Jump Systems (MJSs) with unknown input and measurement noise.It can decouple the unknown input and measurement noise simultaneously.The sufficient conditions for the existence of the observer are given in the form of Linear Matrix Inequality (LMI) to ensure the finite-time stochastic stability of the error system.Thenthe estimation of unknown input is realized by using the idea of algebraic reconstruction.Finallya numerical simulation verifies the effectiveness of the proposed method.