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arXiv · 2611.07138

An extremal unimodular lattice of rank 37

Eric Kubischta
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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 $\mathbb{Z}/13\mathbb{Z}$, 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 $\mathrm{AGL}(1,13)\times C_6$ 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 $\mathbb{Z}/4\mathbb{Z}$-code of length 37 and minimum Euclidean weight 16, resolving the corresponding open case in coding theory.

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Eric Kubischta. 2026-11-04. An extremal unimodular lattice of rank 37. https://arxiv.org/abs/2611.07138

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