An updated table of extremal modular lattices has been established by Michael Juergens during his dissertation project: Extremal lattices
Keywords: modular lattices, tables, minimal norm, 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).
A modular lattice is an integral lattice which is similar to its dual. If the similarity factor is N, the lattice is called N-modular. For example, the root lattice A2 is 3-modular.
If a lattice is N-modular its entry should indicate this by saying:
%PROPERTIES
INTEGRAL=1
MODULAR=N
Some of these entries have an additional field, %SIMILARITY, which contains an integral matrix S such that
The first condition says that S is a similarity of L that multiplies norms by N; the second says that S takes L* to L. Thus S exists if and only if the lattice is N-modular.
See the footnotes file for references.
The table gives the highest possible minimal norm
of a strongly N-modular (even or odd) lattice.
The column headed "+" refers to lattices that are
rationally equivalent to the direct sum of the
appropriate number of copies of the two-dimensional lattice
with Gram matrix
( 1 0 )
( 0 N )
while the column headed "-" refers to the other genera of lattices.
See the footnotes file for references. Any unmarked upper bounds are from [RaSl98]. "NA" means "Not Applicable", or in other words no lattice is possible.
This table is based on the work of many people, including C. Bachoc, G. Nebe, H.-G. Quebbemann, E. M. Rains, R Scharlau, N. J. A. Sloane, B. B. Venkov
Many gaps remain! Please send any improvements to (nebe@math.rwth-aachen.de). Also, we need explicit Gram matrices for the lattices indicated by the footnotes B, C and Q!
| Dim | N = 2 | N = 3 | N = 5 | N = 6 | N = 7 | N = 11 | N = 14 | N = 15 |
|---|---|---|---|---|---|---|---|---|
| - + | - + | - + | - + | - + | - + | |||
| 2 | 1b | 2 1b | 2b 1b | NA | 2b | 2b 3b | 3b NA | NA |
| 4 | 2 | 1d 2d | 2 2d | 2 2 | 3 | 2d 4 | NA 4 | 4 4 |
| 6 | 1S | 2d 2 | 2d 3 | NA | 4 | 4 4 | 4 NA | NA |
| 8 | 2d | 2d 2d | 2d 4 | 2d 4 | 4B | 4 6 | NA 6 | 4d 6 |
| 10 | 2 | 2d 3 | 3 3S | NA | 4B | 6 -6 | -6 NA | NA |
| 12 | 2d | 2d 4 | 4 -4 | 4 -4 | -5g | 6 -7 | NA 8 | -7 8 |
| 14 | 2d | 4 3C | -4 -4 | NA | 6 | 7-8 6-8 | -8 NA | NA |
| 16 | 4 | 2-3 4 | -4 6 | -4 6 | 6 | 8 -9S | NA -10 | -8 10 |
| 18 | 2-3S | 4B 4C | -5 -5S | NA | 6 | -10 -9 | ? NA | NA |
| 20 | 4, 4, 4 | 2-4 4B | 6 -6 | -6 -6 | 8 | -10 -10 | NA -12 | -10 -12 |
| 22 | 2-4 | 4 4C | -6 -6 | NA | -8 | -12 -11 | ? NA | NA |
| 24 | 4 | 3-4 6 | -6 8 | -6 8 | 8-10 | 12-12 | NA 8-14 | -12 -14 |
| 26 | 2-4 | 6N,B 4C-5S | -7 -8 | NA | -10 | -12 -12 | ? NA | NA |
| 28 | 4 | 3-5S 6N,B | 8 -8 | -8 -8 | -10 | -13 -16 | NA -16 | ? -16 |
| 30 | 2-4 | 6 4C-6 | -8 -8 | NA | -11 | -14 -16 | ? NA | NA |
| 32 | 6Q,B | 3-6 6B | -8 -10 | -12 | -15 -18 | NA -18 | ||
| 34 | -5 | 6B 4C-6 | -9 -10 | -12 | ||||
| 36 | 6B | 3-6 6-8 | -10 -10 | -12 | ||||
| 38 | -6 | 4-8 4-8 | -10 -10 | -14 | ||||
| 40 | 6B | -7 8 | -10 -12 | -14 | ||||
| 42 | -6 | -8 -8 | -11 -12 | -14 | ||||
| 44 | 6B | -8 -8 | -12 -12 | -15 | ||||
| 46 | -6 | -8 -8 | -12 -12 | -16 | ||||
| 48 | 8B | -8 8 -10 | -12 12-14 | -16 | 18- | |||
| 50 | ||||||||
| 52 | ||||||||
| 54 | ||||||||
| 56 | ||||||||
| 58 | ||||||||
| 60 | ||||||||
| 62 | ||||||||
| 64 | 12 | |||||||
| 72 | 16-20 | 24-? |