负可定向序列周期的新上界
New upper bounds for the period of a negative orientable sequence
AI总结：
本文提出负可定向序列周期的新上界，改进了现有上界，并指出对于n>2仍存在差距。
摘要：
负可定向序列，即具有有限字母表（至少三个元素）的周期序列，其中n元组或其反向的负数在一个周期中至多出现一次，由Alhakim等人于2024年引入。定义它们的主要目的是作为生成可定向序列的手段，可定向序列有自动位置定位应用，尽管它们本身也可能有自身价值。在本文中，我们开发了负可定向序列周期的新上界，对于n>2的情况，这些上界比已知的上界要显著更紧。开发新上界的方法涉及检查负可定向序列对应的德布鲁因图的子图中的节点，并考虑该子图中每个顶点的入度必须等于出度的含义。然而，尽管改进了上界，对于每个n>2，负可定向序列已知的最大周期与相应上界之间仍存在差距。
英文摘要：
Negative orientable sequences, i.e. periodic sequences with elements from a finite alphabet of size at least three in which an n-tuple or the negative of its reverse appears at most once in a period of the sequence, were introduced by Alhakim et al. in 2024. The main goal in defining them was as a means of generating orientable sequences, which have automatic position location applications, although they are potentially of interest in their own right. In this paper we develop new upper bounds on the period of negative orientable sequences which, for n>2, are significantly sharper than the previous known bound. The approach used to develop the new bounds involves examining the nodes in the subgraph of the de Bruijn graph corresponding to a negative orientable sequence, and to consider the implications of the fact that the in-degree of every vertex in this subgraph must equal the out-degree. However, despite improving the bounds, a gap remains between the largest known period for a negative orientable sequence and the corresponding bounds for every n>2.
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