Unique Form and Some Construction of Uncorrelated Optimal ZCZ Sequences

In this paper, we prove that (Theorem 1) any uncorrelated optimal Zero-Correlation-Zone (ZCZ) sequence family comes from the Popović's construction, and (Theorem 2) propose a construction for uncorrelated optimal families of ZCZ sequences over a polyphase alphabet. Additionally, we prove that (Theorems 3 and 4) any polyphase perfect sequence comes from Mow's unified construction if and only if any polyphase uncorrelated optimal ZCZ sequence family comes from our proposed construction. In the meantime, we prove some new interesting properties (Propositions 1, 2 and 3) of optimal ZCZ sequence families. The first corollary proves that the alphabet size is lower bounded by K when K is the number of optimal ZCZ sequences in the family. The second corollary proves that the interleaved sequence of an uncorrelated optimal ZCZ sequence family becomes a perfect sequence, and we conjecture that the reverse operation works when the ZCZ width is 1. From the second corollary and under the assumption that Mow's conjecture on the alphabet size of perfect sequences holds, we derive (Remark 2) the minimum alphabet size for uncorrelated optimal families of ZCZ sequences over a polyphase alphabet. Furthermore, we provide (Corollary 3) a simple explicit form of the uncorrelated optimal families of ZCZ sequences with the minimum alphabet size.