In this paper, we give a construction of optimal families of $N$ -ary perfect sequences of period $N^{2}$ , where $N$ is a positive odd integer. For this, we re-define perfect generators and optimal generators of any length $N$ which were originally defined only for odd prime lengths by Park, Song, Kim, and Golomb in 2016, but investigate the necessary and sufficient condition for these generators for arbitrary length $N$ . Based on this, we propose a construction of odd length optimal generators by using odd prime length optimal generators. For a fixed odd integer $N$ and its odd prime factor $p$ , the proposed construction guarantees at least $(N/p)^{p-1}\phi (N/p)\phi (p)\phi (p-1)/\phi (N)^{2}$ inequivalent optimal generators of length $N$ in the sense of constant multiples, cyclic shifts, and/or decimations. Here, $\phi (\cdot )$ is Euler’s totient function. From an optimal generator one can construct lots of different $N$ -ary optimal families of period $N^{2}$ , all of which contain $p_{\text {min}}-1$ perfect sequences, where $p_{\text {min}}$ is the least positive prime factor of $N$ .