A universal partial cycle (or upcycle) for $\mathcal{A}^n$ is a cyclic sequence that covers each word of length $n$ over the alphabet $\mathcal{A}$ exactly once -- like a De Bruijn cycle, except that we also allow a wildcard symbol $\mathord{\diamond}$ that can represent any letter of $\mathcal{A}$. Chen et al. in 2017 and Goeckner et al. in 2018 showed that the existence and structure of upcycles are highly constrained, unlike those of De Bruijn cycles, which exist for any alphabet size and word length. Moreover, it was not known whether any upcycles existed for $n \ge 5$. We present several examples of upcycles over both binary and non-binary alphabets for $n = 8$. We generalize two graph-theoretic representations of De Bruijn cycles to upcycles. We then introduce novel approaches to constructing new upcycles from old ones. Notably, given any upcycle for an alphabet of size $a$, we show how to construct an upcycle for an alphabet of size $ak$ for any $k \in \mathbb{N}$, so each example generates an infinite family of upcycles. We also define folds and lifts of upcycles, which relate upcycles with differing densities of $\mathord{\diamond}$ characters. In particular, we show that every upcycle lifts to a De Bruijn cycle. Our constructions rely on a different generalization of De Bruijn cycles known as perfect necklaces, and we introduce several new examples of perfect necklaces. We extend the definitions of certain pseudorandomness properties to partial words and determine which are satisfied by all upcycles, then draw a conclusion about linear feedback shift registers. Finally, we prove new nonexistence results based on the word length $n$, alphabet size, and $\mathord{\diamond}$ density.

翻译：通用偏循环（或称上循环）是定义在字母表$\mathcal{A}$上的一个循环序列，它精确覆盖每个长度为$n$的单词一次——类似于德布鲁因循环，但允许通配符$\mathord{\diamond}$表示$\mathcal{A}$中的任意字母。Chen等人（2017年）与Goeckner等人（2018年）的研究表明，上循环的存在性和结构受到严格限制，这与可适用于任意字母表大小和单词长度的德布鲁因循环不同。此外，对于$n \ge 5$的情形，此前未知是否存在任何上循环。本文给出了$n=8$时在二进制及非二进制字母表上的多个上循环实例，并将德布鲁因循环的两种图论表示推广至上循环。我们随后提出从旧上循环构建新上循环的创新方法。特别地，对于任意大小为$a$的字母表上的上循环，我们展示了如何为任意$k \in \mathbb{N}$构造大小为$ak$的字母表上的上循环，因此每个实例均可生成一个无限上循环族。我们还定义了上循环的折叠与提升，用以关联含不同密度$\mathord{\diamond}$字符的上循环。具体而言，我们证明每个上循环均可提升为德布鲁因循环。我们的构造依赖于德布鲁因循环的另一种推广——完美项链，并引入了多个新型完美项链实例。我们将某些伪随机性属性的定义扩展至偏词，并确定所有上循环满足的属性，进而得出关于线性反馈移位寄存器的结论。最后，基于单词长度$n$、字母表大小及$\mathord{\diamond}$密度，我们证明了新的不存在性结论。