In direct-sequence code-division multiple-access (DS-CDMA) communication systems and direct-sequence ultra wideband (DS-UWB) radios, sequences with low correlation and large family size are important for reducing multiple access interference (MAI) and accepting more active users, respectively. In this paper, a new collection of families of sequences of length pⁿ-1, which includes three constructions, is proposed. The maximum number of cyclically distinct families without GMW sequences in each construction is φ(pⁿ-1)/n·φ(p^m-1)/m, where p is a prime number, n is an even number, and n=2m, and these sequences can be binary or polyphase depending upon choice of the parameter p. In Construction I, there are pⁿ distinct sequences within each family and the new sequences have at most d+2 nontrivial periodic correlation {-p^m-1, -1, p^m-1, 2p^m-1,…,dp^m-1}. In Construction II, the new sequences have large family size p²ⁿ and possibly take the nontrivial correlation values in {-p^m-1, -1, p^m-1, 2p^m-1,…,(3d-4)p^m-1}. In Construction III, the new sequences possess the largest family size p^(d-1)n and have at most 2d correlation levels {-p^m-1, -1,p^m-1, 2p^m-1,…,(2d-2)p^m-1}. Three constructions are near-optimal with respect to the Welch bound because the values of their Welch-Ratios are moderate, WR_??_d, WR_??_3d-4 and WR_??_2d-2, respectively. Each family in Constructions I, II and III contains a GMW sequence. In addition, Helleseth sequences and Niho sequences are special cases in Constructions I and III, and their restriction conditions to the integers m and n, p^m≠2 (mod 3) and n≅0 (mod 4), respectively, are removed in our sequences. Our sequences in Construction III include the sequences with Niho type decimation 3·2^m-2, too. Finally, some open questions are pointed out and an example that illustrates the performance of these sequences is given.