### A Conjecture on Hermite Constants

##### Abstract

As of today, the Hermite constants $\gamma_n$ are only known for $n\in \{1,2,3,4,5,6,7,8,24\}$. We noted that the known values of $(4/\gamma_n)^n$ coincide with the values of the minimal determinants of any $n$-dimensional integral lattice when the length of the smallest lattice element $\mu$ is fixed to 4. Based on this observation, we conjecture that the values of $\gamma_n^n$ for $n=9,\ldots,23$ are those given in Table 2. We provide a supporting argument to back this conjecture. We also provide a provable lower bound on the Hermite constants for $1\leq n\leq24$.

Available format(s)
Category
Foundations
Publication info
Preprint.
Keywords
Hermite constants lattices conjectures
Contact author(s)
leon-philipp machler @ ens fr
david naccache @ ens fr
History
2022-06-13: last of 15 revisions
See all versions
Short URL
https://ia.cr/2022/677

CC BY

BibTeX

@misc{cryptoeprint:2022/677,
author = {Leon Mächler and David Naccache},
title = {A Conjecture on Hermite Constants},
howpublished = {Cryptology ePrint Archive, Paper 2022/677},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/677}},
url = {https://eprint.iacr.org/2022/677}
}

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