Table of Strongly Perfect Lattices

Keywords: tables, perfect lattices, quadratic forms

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, RWTH Aachen university (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, (njasloane@gmail.com).

Last modified Mai 2019

All known Strongly Perfect Lattices up to dimension 26

This table of strongly perfect lattices is complete through 13 dimensions, in dimension 14,15,16 it is complete assuming that also the dual lattice is strongly perfect.

1 dimension SP1.1 = A1

2 dimensions SP2.1 = A2

4 dimensions SP4.1 = D4,

6 dimensions SP6.1 = E6, SP6.2 = E6*,

7 dimensions SP7.1 = E7, SP7.2 = E7*,

8 dimensions SP8.1 = E8

10 dimensions SP10.1 = K10', SP10.2 = K10'*

12 dimensions SP12.1 = K12=Coxeter Todd lattice

14 dimensions SP14.1 = Q14 = extremal 3-modular lattice

16 dimensions SP16.1 = Lambda16 = Barnes Wall lattice, SP16.2 = N16 = extremal 5-modular lattice, SP16.3 = O16 = over Barnes Wall lattice, SP16.4 = O16*, SP16.5 = Gamma16 , SP16.6 = Gamma16*

18 dimensions KAPPA18, KAPPA18*

20 dimensions The three extremal 2-modular lattices (SU(5,2) x SL(2,3)).C2, C2.M12.C2, HS20

21 dimensions KAPPA21

22 dimensions LAMBDA22, LAMBDA22*, LAMBDA22*2, LAMBDA22*3, O22, O22*, M22, M22*, M22*3, M22*5

23 dimensions LAMBDA23, LAMBDA23*, the shorter Leech lattice O23, M23, M23*, M23*2, M23*3

24 dimensions Leech lattice, L_24.2

26 dimensions The two known extremal 3-modular lattices S6(3)C3.2, unimodular Eisenstein lattice

Remarks

The table is based on the one in the paper by Boris Venkov, Reseaux and designs spheriques. There are two additional entries in dimension 16, discovered by Sihuang Hu and Gabriele Nebe. By a result by Christine Bachoc and Boris Venkov

all extremal even unimodular lattices in dimensions congruent to 0 or 8 mod 24 are strongly perfect,

all extremal even 2-modular lattices in dimensions congruent to 0 or 4 mod 16 are strongly perfect and

all extremal even 3-modular lattices in dimensions congruent to 0 or 2 mod 12 are strongly perfect.

References

Boris Venkov, Reseaux and designs spheriques. L'Enseignement Mathematique, Geneve 2001
Gabriele Nebe, Boris Venkov, The strongly perfect lattices of dimension 10. Colloque International de Theorie des Nombres (Talence, 1999). J. Theor. Nombres Bordeaux 12 (2000), no. 2, 503-518.
Gabriele Nebe, Boris Venkov, Low-dimensional strongly perfect lattices. I. The 12-dimensional case. Enseign. Math. (2) 51 (2005), no. 1-2, 1290163.
Gabriele Nebe, Elisabeth Nossek, Boris Venkov, Low dimensional strongly perfect lattices. II: Dual strongly perfect lattices of dimension 13 and 15. J. Theor. Nombres Bordeaux 25 (2013), no. 1, 147-161.
Gabriele Nebe, Boris Venkov, Low-dimensional strongly perfect lattices. III. Dual strongly perfect lattices of dimension 14. Int. J. Number Theory 6 (2010), no. 2, 387-409.
Sihuang Hu, Gabriele Nebe, Strongly perfect lattices sandwiched between Barnes-Wall lattices
Sihuang Hu, Gabriele Nebe, Low-dimensional strongly perfect lattices. IV. Dual strongly perfect lattices of dimension 16.
J. Martinet, Les R'eseaux Parfaits des Espaces Euclidiens, Masson, Paris, 1996.
J. Martinet, Home page (among other things, lists all known strongly perfect lattices up to dimension 26).

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