Certain Important Lattices and Their Properties

  • J. H. Conway
  • N. J. A. Sloane
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 290)


This chapter describes the properties of a number of important lattices, including the cubic lattice Z n , the A n , D n , E 6, E 7, E 8, the Coxeter-Todd lattice K 12, the Barnes-Wall lattice Λ16, the Leech lattice Λ24, and their duals. Among other things we give their minimal vectors, densities, covering radii, glue vectors, automorphism groups, expressions for their theta series, and tables of the numbers of points in the first fifty shells. We also include a brief discussion of reflection groups and of the technique of gluing lattices together.


Root Lattice Automorphism Group Generator Matrix Weyl Group Minimal Norm 


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  1. Adol] J.-P. Adoul, La quantification vectorielle des signaux: approche algebrique,Ann. Te’écommun. 41 (1986), 158–177 [2, 20].Google Scholar

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© Springer Science+Business Media New York 1999

Authors and Affiliations

  • J. H. Conway
  • N. J. A. Sloane

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