On a generalization of Craig lattices

Hao Chen[1]

  • [1] Software Engineering Institute East China Normal University Zhong Shan North Road 3663 Shanghai 200062, P.R. China

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 1, page 59-70
  • ISSN: 1246-7405

Abstract

top
In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range 3332 - 4096 which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range 128 - 3272 . We also construct some dense lattices of dimensions in the range 4098 - 8232 . Finally we also obtain some new lattices of moderate dimensions such as 68 , 84 , 85 , 86 , which are denser than the previously known densest lattices.

How to cite

top

Chen, Hao. "On a generalization of Craig lattices." Journal de Théorie des Nombres de Bordeaux 25.1 (2013): 59-70. <http://eudml.org/doc/275737>.

@article{Chen2013,
abstract = {In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68, 84, 85, 86$, which are denser than the previously known densest lattices.},
affiliation = {Software Engineering Institute East China Normal University Zhong Shan North Road 3663 Shanghai 200062, P.R. China},
author = {Chen, Hao},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {generalized Craig lattices},
language = {eng},
month = {4},
number = {1},
pages = {59-70},
publisher = {Société Arithmétique de Bordeaux},
title = {On a generalization of Craig lattices},
url = {http://eudml.org/doc/275737},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Chen, Hao
TI - On a generalization of Craig lattices
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 59
EP - 70
AB - In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68, 84, 85, 86$, which are denser than the previously known densest lattices.
LA - eng
KW - generalized Craig lattices
UR - http://eudml.org/doc/275737
ER -

References

top
  1. R. Bacher, Dense lattices in dimensions 27-29. Invent. Math. 130 (1997), 153–158. Zbl0879.11034MR1471888
  2. A. Bos, J. H. Conway and N. J. A. Sloane, Further lattices packings in high dimensions. Mathematika 29 (1982), 171–180. Zbl0489.52017MR696873
  3. J. Carmelo Interlando, A. L. Flores and T. P. da Nóbrega Neto, A family of asymptotically good lattices having a lattice in each dimension. Inter. J. Number Theory 4 (2008), 147–154. Zbl1200.11047MR2387922
  4. H. Cohn and N. D. Elkies, New upper bounds on sphere packings I. Ann. of Math. 157 (2003), 689–714. Zbl1041.52011MR1973059
  5. H. Cohn and A. Kumar, Optimality and uniqueness of Leech lattice among lattices. Ann. of Math 170(2009), 1003–1050. Zbl1213.11144MR2600869
  6. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups. 3rd Edition, Springer, 1999. Zbl0915.52003MR1662447
  7. J. H. Conway and N. J. A. Sloane, Laminated lattices. Ann. of Math. 116 (1982), 593–620. Zbl0502.52016MR678483
  8. J. H. Conway and N. J. A. Sloane, The antipode construction for sphere packings. Invent. Math. 123 (1996), 309–313. Zbl0852.52010MR1374202
  9. J. H. Conway, Sphere packings, lattices, codes and greed. Proc. ICM Zurich, I (1994), 45–55. Zbl0853.52016MR1403914
  10. M. Craig, Extreme forms and cyclotomy. Mathematika 25 (1978), 44–56. Zbl0395.10038MR491524
  11. M. Craig, A cyclotomic construction for Leech’s lattice. Mathematika 25 (1978), 236–241. Zbl0413.10024MR533130
  12. N. D. Elkies, Mordell-Weil lattices in characteristic 2, I: Construction and first properties. Inter. Math. Research Notices 8 (1994), 343–361. Zbl0813.52017MR1289579
  13. N. D. Elkies, Mordell-Weil lattices in characteristic 2, II: The Leech lattice as a Mordell-Weil lattice. Invent. Math. 128 (1997),1–8. Zbl0897.11023MR1437492
  14. N. D. Elkies, Mordell-Weil lattices in characteristic 2, III: A Mordell-Weil lattice of rank 128. Experimental Math. 10 (2001), 467–473. Zbl1040.11041MR1917431
  15. A. L. Flores, J. Carmelo Interlando, T. P. da N. Neto and J. O. D. Lopos, On a refinement of Craig’s lattice. J. Pure Appl. Algebra 215 (2011), 1440–1442. Zbl1228.11094MR2769242
  16. C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternaren quadratischen Formen von Ludwig August Seeber. Göttingische gelehrte Anzeigen 1831, Juli 9, = “Recension der...” in J. Reine Angew Math. 20, 312–320, = Werke 11 (1840), 188–196. 
  17. M. Grassl, http://www.codetables.de 
  18. T. C. Hales, A proof of the Kepler conjecture. Ann. of Math. 162 (2005), 1065–1185. Zbl1096.52010MR2179728
  19. J. Kepler, The six-cornered snowflake (1611); translated by L. L. Whyte, Oxford Univ. Press, 1966. 
  20. A. Korkine and G. Zolotareff, Sur les formes quadratiques positives. Math. Ann. 11 (1877), 242–292. MR1509914
  21. F. R. Kschischang and S. Pasupathy, Some ternary and quaternary codes and associated sphere packings. IEEE Trans. on Information Theory 38 (1992), 227–246. Zbl0744.94024MR1162203
  22. J. Leech, Notes on sphere packings. Canadian J. Math. 19 (1967), 251–267. Zbl0162.25901MR209983
  23. J. Leech and N. J. A. Sloane, New sphere packings in dimension 9 - 15 . Bull. Amer. Math. Soc. 76 (1970), 1006–1010. Zbl0198.55103MR266056
  24. G. Nebe and N. J. A. Sloane, A Catalogue of Lattices. http://www.math.rwth-aachen.de/ Gabriele.Nebe/LATTICES/ 
  25. H.-G. Quebbemann, A Construction of integral lattices. Mathematika 31 (1984), 137–140. Zbl0538.10028MR762185
  26. A. Schürmann, Computational geometry of positive quadratic forms: polyhedral reduction theory, algorithms and applications. University Lecture Series, Amer. Math. Soc. 2008. 
  27. I. Shimada, Lattices of algebraic cycles on Fermat varieties in positive characteristics. Proc. London Math. Soc. 82 (2001), 131–172. Zbl1074.14508MR1794260
  28. T. Shioda, Mordell-Weil lattices and sphere packings. Amer. J. Math. 113 (1991), 931–948. Zbl0756.14010MR1129298
  29. N. J. A. Sloane, The sphere packing problem. Proc. ICM III, Berlin (1998), 387–396. Zbl0906.11034MR1648172
  30. J. H. van Lint, Introduction to coding theory. 3rd Edition, Springer-Verlag, 1999. Zbl0747.94018MR1664228

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.