构造更长的蛇并改进超立方体中的渐近界
Constructing longer snakes and improved asymptotic bounds in hypercubes
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中文总结 AI 辅助
本研究构造了维度13至20中更长的蛇，并改进$d\geq21$的下界至$(17/48)2^d$，通过兼容路径和受控重叠实现，且可独立验证。
中文摘要 AI 辅助
我们给出了在维度13至20中比先前已知更长的蛇，并改进了每个维度$d \geq 21$的一般下界。我们的显式蛇在维度20中达到371,711条边。该立方体中的二十条兼容路径允许推广，从而对每个$d \geq 21$给出长度至少为$(17/48)2^d$的蛇。它们的受控重叠允许副本在更大立方体的层间连接而不产生捷径。四条额外的路径为线圈给出相同的界。我们解释构造，证明连接规则，然后计算其长度。有限路径及其所需交集可独立验证。
英文摘要
We give snakes that are longer than the previous best known in dimensions 13 through 20 and improve the general lower bound for every dimension $d \geq 21$. Our explicit snakes reach 371,711 edges in dimension 20. Twenty compatible paths in that cube allow generalisation to give snakes of length at least $(17/48)2^d$ for every $d \geq 21$. Their controlled overlaps allow copies to be joined across the layers of a larger cube without creating shortcuts. Four additional paths give the same bound for coils. We explain the construction, prove the joining rule, and then count its length. The finite paths and their required intersections are independently verifiable.
