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In this paper, we propose three classes of systematic approaches for constructing zero-correlation zone (ZCZ) sequence families. In most cases, these approaches are capable of generating sequence families that achieve the upper bounds on the family size and the ZCZ width for a given sequence period. Our approaches can produce various binary and polyphase ZCZ families with desired parameters and alphabet size. They also provide additional tradeoffs amongst the above four system parameters and are less constrained by the alphabet size. Furthermore, the constructed families have nested-like property that can be either decomposed or combined to constitute smaller or larger ZCZ sequence sets. We make detailed comparisons with related works and present some extended properties. For each approach, we provide examples to numerically illustrate the proposed construction procedure.