This paper generalizes a method of generating shift sequences in the interleaved construction proposed by Gong.With the new shift sequences,some new families of p-ary sequences with desired properties can be obtained.A lower bound on the number of new families of binary sequences is also established. This paper generalizes a method of generating shift sequences in the interleaved construction proposed by Gong.With the new shift sequences, some new families of p-ary sequences with desired properties can be obtained. A lower bound on the number of new families of binary sequences is also established.