A note on Quebbemann’s extremal lattices of rank 64
Ichiro Shimada1
1 Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima, 739-8526, Japan
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 813-826.

By constructing explicit examples, we show that the method of Quebbemann yields many isomorphism classes of extremal lattices of rank 64. Many of these examples have no non-trivial automorphisms.

En construisant des exemples explicites, nous montrons que la méthode de Quebbemann permet d’obtenir de nombreuses classes d’isomorphisme de réseaux extrémaux de dimension 64. Beaucoup de ces exemples n’ont pas d’automorphismes non triviaux.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1229
Classification: 11H31, 11H56
Keywords: extremal lattice

Ichiro Shimada 1

1 Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima, 739-8526, Japan
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Ichiro Shimada. A note on Quebbemann’s extremal lattices of rank $64$. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 813-826. doi : 10.5802/jtnb.1229. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1229/

[1] Etsuko Bannai Positive definite unimodular lattices with trivial automorphism groups, Memoirs of the American Mathematical Society, 429, American Mathematical Society, 1990, 70 pages | Zbl

[2] John H. Conway; Neil J. A. Sloane Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, 290, Springer, 1999, lxxiv+703 pages | DOI

[3] The GAP Group GAP — Groups, Algorithms, and Programming, Version 4.11.0, 2020 (http://www.gap-system.org)

[4] Masaaki Harada; Masaaki Kitazume; Michio Ozeki Ternary code construction of unimodular lattices and self-dual codes over ℤ 6 , J. Algebr. Comb., Volume 16 (2002) no. 2, pp. 209-223 | DOI | Zbl

[5] Masaaki Harada; Tsuyoshi Miezaki On the existence of extremal Type II ℤ 2k -codes, Math. Comput., Volume 83 (2014) no. 287, pp. 1427-1446 | DOI | Zbl

[6] Y. Mimura A construction of unimodular lattices without non-trivial automorphisms (a survey), RIMS Kokyuroku, Volume 1476 (2006), pp. 105-110

[7] Gabriele Nebe Some cyclo-quaternionic lattices, J. Algebra, Volume 199 (1998) no. 2, pp. 472-498 | DOI | Zbl

[8] Meinhard Peters Definite unimodular 48-dimensional quadratic forms, Bull. Lond. Math. Soc., Volume 15 (1983), pp. 18-20 | DOI | Zbl

[9] Heinz-Georg Quebbemann An application of Siegel’s formula over quaternion orders, Mathematika, Volume 31 (1984), pp. 12-16 | DOI | Zbl

[10] Heinz-Georg Quebbemann A construction of integral lattices, Mathematika, Volume 31 (1984), pp. 137-140 | DOI

[11] Ichiro Shimada An even extremal lattice of rank 64, J. Number Theory, Volume 185 (2018), pp. 1-15 | DOI | Zbl

[12] Ichiro Shimada A note on Quebbemann’s extremal lattices of rank 64: computation data, 2021 (http://www.math.sci.hiroshima-u.ac.jp/shimada/lattice.html)

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