We consider a general framework for constructing non-linear generators by adding a (32-bit or larger) pseudo-random number generator (PRNG) as a baseline generator to the basic RC4 design, in which an index-selection scheme similar to RC4 is used. We refer to the proposed design as the eRC (enhanced/extended RC4) design. We discuss several advantages of adding a good baseline generator to the RC4 design, including new updating schemes for the auxiliary table. We consider some popular PRNGs with the nice properties of high-dimensional equi-distribution, efficiency, long period, and portability as the baseline generator. We demonstrate that eRC generators are very efficient via extensive empirical testing on some eRC generators. We also show that eRC is flexible enough to choose minimal design parameters for eRC generators and yet the resulting eRC generators still pass stringent empirical tests, which makes them suitable for both software and hardware implementations.