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An extremal unimodular lattice of rank 37
Abstract
We construct an extremal unimodular lattice of rank 37, resolving the last undetermined optimal minimum in ranks at most 48. The construction begins with the Cayley table of ℤ/13ℤ, assigning a vector of norm four to each cell. We prove that the resulting incidence lattice has exactly 26 unimodular completions of minimum four, all isometric. The resulting Cayley lattice Λ has automorphism group AGL(1,13)× C6 and is s-extremal, with shadow minimum 13/4 and kissing number 35 594. An explicit $4$-frame in Λ yields a Type I self-dual ℤ/4ℤ-code of length 37 and minimum Euclidean weight 16, resolving the corresponding open case in coding theory.
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BibTeX
@misc{kubischta2026extremal,
title = {{An extremal unimodular lattice of rank 37}},
author = {Eric Kubischta},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07138},
eprint = {2611.07138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07138},
}
RIS
TY - UNPB
TI - An extremal unimodular lattice of rank 37
AU - Eric Kubischta
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07138
UR - https://arxiv.org/abs/2611.07138
AB - We construct an extremal unimodular lattice of rank 37, resolving the last undetermined optimal minimum in ranks at most 48. The construction begins with the Cayley table of ℤ/13ℤ, assigning a vector of norm four to each cell. We prove that the resulting incidence lattice has exactly 26 unimodular completions of minimum four, all isometric. The resulting Cayley lattice Λ has automorphism group AGL(1,13)× C_6 and is s-extremal, with shadow minimum 13/4 and kissing number 35 594. An explicit $4$-frame in Λ yields a Type I self-dual ℤ/4ℤ-code of length 37 and minimum Euclidean weight 16, resolving the corresponding open case in coding theory.
ER -