A Construction of Odd Length Generators for Optimal Families of Perfect Sequences

In this paper, we give a construction of optimal families of N-ary perfect sequences of period N², 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-1φ(N/p)φ(p)φ(p-1)/φ(N)² inequivalent optimal generators of length N in the sense of constant multiples, cyclic shifts, and/or decimations. Here, φ (·) is Euler's totient function. From an optimal generator one can construct lots of different N-ary optimal families of period N², all of which contain p_min^-1 perfect sequences, where p_min is the least positive prime factor of N.