We propose a general class of uniform random variate generators with the nice property of High-dimensional uniformity, Efficiency, Long cycle and Portability (HELP). It includes FMRG-k (Fast Multiple Recursive Generator), proposed by Deng and Lin (2000), as a special class. FMRG-k, belonging to a wider class Multiple Recursive Generator (MRG), is a much more efficient generator than a general MRG. Here k is the degree of polynomial used in the recurrence relations. The value of k will determine the dimension of uniformity and the period length for MRG (and FMRG-k. In general, we prefer to find a larger value of k for FMRG-k because larger k will improve the quality of the generator but it will not increase the computing time. We will first extend FMRG-k to a wider class, called HELP-k, for a large k. An efficient search algorithm will be discussed. In particular, we currently extend results by Deng and Lin (2000) for FMRG-k, from k <= 4 to some k > 100. Listing of HELP-k and its C code are also given. We offer HELP-k as an alternative to MT19937, a powerful random number generator proposed by Matumoto and Nishimura (1998). More comparisons will be given.