关于变长非重叠码的最大规模
On the maximum size of variable-length non-overlapping codes
AI总结：
本文通过建立变长与定长非重叠码的联系，证明长度至多 n 的 q 元变长非重叠码规模以 C(n,q) 为上界，并给出码字平均长度下界及渐近结果。
摘要：
非重叠码是一组码字，其中每个码字的前缀都不是该集合中任何码字（包括其自身）的后缀。如果码字长度可变，还额外要求每个码字都不能作为子词包含在任何其他码字中。令 C(n,q) 表示长度为 n 的 q 元定长非重叠码的最大规模。关于 C(n,q) 的上界已有充分研究。然而，长度至多为 n 的变长非重叠码最大规模的非平凡上界仍未解决。本文通过建立变长非重叠码与定长非重叠码之间的联系，证明了 q 元变长非重叠码的规模以 C(n,q) 为上界。此外，我们证明了 q 元变长非重叠码中码字的平均长度以 ⌈log_q \ ilde{C}⌉ 为下界，并且当 q 趋于无穷时，其渐近长度不短于 n−2，其中 \ ilde{C} 表示长度至多为 n 的 q 元变长非重叠码的基数。
英文摘要：
Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let $C(n,q)$ be the maximum size of $q$-ary fixed-length non-overlapping codes of length $n$. The upper bound on $C(n,q)$ has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes of length at most $n$ remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a $q$-ary variable-length non-overlapping code is upper bounded by $C(n,q)$. Furthermore, we prove that the average length of the codewords in a $q$-ary variable-length non-overlapping codes is lower bounded by $\lceil \log_q \tilde{C} \rceil$, and is asymptotically no shorter than $n-2$ as $q$ approaches $\infty$, where $\tilde{C}$ denotes the cardinality of $q$-ary variable-length non-overlapping codes of length up to $n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
