PreprintCC BY 4.0arXiv
Constructing longer snakes and improved asymptotic bounds in hypercubes
Abstract
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 ≥ 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)2d for every d ≥ 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.
Related
BibTeX
@misc{taylor2026constructing,
title = {{Constructing longer snakes and improved asymptotic bounds in hypercubes}},
author = {Tom Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27483},
eprint = {2609.27483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27483},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Constructing longer snakes and improved asymptotic bounds in hypercubes
AU - Tom Taylor
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27483
UR - https://arxiv.org/abs/2609.27483
AB - 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 ≥ 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 ≥ 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.
ER -