A perfect secret sharing scheme allows a secret K to be shared among a set of participants in such a way that only qualified subsets of participants can recover the secret, and unqualified subsets of participants obtain no information regarding the secret. In this paper, we propose a construction of perfect secret sharing schemes with uniform, generalized access structures of rank 3. Compared with other constructions, our construction has some improved lower bounds on the information rate. In addition, we also generalize the construction to perfect secret sharing schemes with uniform, generalized access structures of constant rank.