In this paper, we introduce a new family of polynomials which generates reversible codes over a finite field with sixteen elements ($F_16$ or $GF(16)$). We name the polynomials in this family as lifted polynomials. Some advantages of lifted polynomials are that they are easy to construct, there are plenty of examples of them and it is easy to determine the dimension of codes generated by them. Furthermore we introduce 4-lifted polynomials which provide a rich source for DNA codes. Also we construct codes over $F_4$ that have the best possible parameters from lifted polynomials. In addition we obtain some reversible codes over $F_4$.