# On the construction of dense lattices with a given automorphisms group

Philippe Gaborit^{[1]}; Gilles Zémor^{[2]}

- [1] Université de Limoges, XLIM 123 av. A. Thomas, 87000 Limoges (France)
- [2] Université Bordeaux I 351 av. de la Libération 33405 Talence (France)

Annales de l’institut Fourier (2007)

- Volume: 57, Issue: 4, page 1051-1062
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topGaborit, Philippe, and Zémor, Gilles. "On the construction of dense lattices with a given automorphisms group." Annales de l’institut Fourier 57.4 (2007): 1051-1062. <http://eudml.org/doc/10250>.

@article{Gaborit2007,

abstract = {We consider the problem of constructing dense lattices in $\mathbb\{R\} ^n$ with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least $cn2^\{-n\}$, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions $n$, we exhibit a finite set of lattices that come with an automorphisms group of size $n$, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp$(n \log n)$, which improves upon previous theorems that yield an equivalent lattice packing density. The method developed here involves applying Leech and Sloane’s Construction A to a special class of codes with a given automorphisms group, namely the class of double circulant codes.},

affiliation = {Université de Limoges, XLIM 123 av. A. Thomas, 87000 Limoges (France); Université Bordeaux I 351 av. de la Libération 33405 Talence (France)},

author = {Gaborit, Philippe, Zémor, Gilles},

journal = {Annales de l’institut Fourier},

keywords = {Lattice packings; Minkowski-Hlawka lower bound; probability; automorphism group; double circulant codes; lattice packing},

language = {eng},

number = {4},

pages = {1051-1062},

publisher = {Association des Annales de l’institut Fourier},

title = {On the construction of dense lattices with a given automorphisms group},

url = {http://eudml.org/doc/10250},

volume = {57},

year = {2007},

}

TY - JOUR

AU - Gaborit, Philippe

AU - Zémor, Gilles

TI - On the construction of dense lattices with a given automorphisms group

JO - Annales de l’institut Fourier

PY - 2007

PB - Association des Annales de l’institut Fourier

VL - 57

IS - 4

SP - 1051

EP - 1062

AB - We consider the problem of constructing dense lattices in $\mathbb{R} ^n$ with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least $cn2^{-n}$, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions $n$, we exhibit a finite set of lattices that come with an automorphisms group of size $n$, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp$(n \log n)$, which improves upon previous theorems that yield an equivalent lattice packing density. The method developed here involves applying Leech and Sloane’s Construction A to a special class of codes with a given automorphisms group, namely the class of double circulant codes.

LA - eng

KW - Lattice packings; Minkowski-Hlawka lower bound; probability; automorphism group; double circulant codes; lattice packing

UR - http://eudml.org/doc/10250

ER -

## References

top- R. Bacher, A new inequality for the Hermite constants, arXiv:math.NT/0603477 (2006) Zbl1165.11056
- K. Ball, A lower bound for the optimal density of lattice packings, Internat. Math. Res. Notices 10 (1992), 217-221 Zbl0776.52006MR1191572
- J. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, 290 (1999), Springer-Verlag, New-York (third edition) Zbl0634.52002MR1662447
- H. Davenport, C. A. Rogers, Hlawka’s theorem in the geometry of numbers, Duke Math. J. 14 (1947), 367-375 Zbl0030.34602
- P. Gaborit, G. Zémor, Asymptotic improvement of the Gilbert-Varshamov bound for linear codes, Inter. Symp. Inf. Theo., ISIT 2006, Seattle (2006), 287-291 Zbl1318.94120
- D. R. Heath-Brown, Zero-free regions for Dirichlet $L$-functions and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), 265-338 Zbl0739.11033MR1143227
- Michael Krivelevich, Simon Litsyn, Alexander Vardy, A lower bound on the density of sphere packings via graph theory, Int. Math. Res. Not. (2004), 2271-2279 Zbl1071.52018MR2076096
- C. A. Rogers, Existence theorems in the geometry of numbers, Ann. of Math. (2) 48 (1947), 994-1002 Zbl0036.02701MR22863
- J. A. Rush, A lower bound on packing density, Invent. Math 98 (1989), 499-509 Zbl0659.10033MR1022304
- J. A. Rush, N. J. A. Sloane, An improvement to the Minkowski-Hlawka bound for packing superballs, Mathematika 34 (1987), 8-18 Zbl0606.10028MR908835
- S. Shlosman, M. Tsfasman, Random lattices and random sphere packings: typical properties, Mosc. Math. J. 1 (2001), 73-89 Zbl0999.11032MR1852935

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.