State estimation for discrete-time linear systems is addressed by developing a filter that provides an estimate of the state depending only on a batch of recent measurement and input vectors. This problem has been solved by introducing a receding-horizon objective function that includes also a weighted penalty term related to the prediction of thestate. Convergence results and unbiasedness properties have been proved for this estimator in a previous work. In this paper, the focus is on the problem of designing such a filter using results related to quadratic boundedness of the estimation error. Upper bounds on the norm of the estimation error have been found by constructing a suitable positively invariant set. Moreover, these bounds may be expressed in terms of Linear Matrix Inequalities (LMIs), and are well-suited to being minimized for the purpose of design.