For the caching problem, when the number of files is no larger than that of users, the best known rate-memory region is achieved by memory sharing between the rate-memory pairs obtained by three schemes: the scheme proposed by Yu et al., the scheme proposed by Gomez-Vilardebo, and the scheme proposed by Tian and Chen. While the first two schemes operate on the binary field, the Tian-Chen scheme makes use of a finite field of order 2(m) with, in some situations, m >= K log(2) (N) for a caching systems with K users and N files. The practical implications of this increase in the size of the field are equivalent to an increase, by a factor of in, in the number of suhfile partitions required. We propose a novel caching scheme that approaches the rate-memory region achieved by the Tian-Chen scheme as the number of users in the system increases, which only requires a field of order 2(2).