This data-base of lattices is a joint project of
Neil J. A. Sloane (email@example.com) and
Gabriele Nebe, RWTH Aachen University (firstname.lastname@example.org)
Our aim is to give information about all the interesting
lattices in "low" dimensions.
Contributions are welcome.
For a list of the lattices in the data-base, see the Index file.
The files in the data-base are in two different formats:
(i) the standard format (with a file name such as E8.std), which is what we use internally; these files are in plain ASCII, which is the most convenient for computations; and
(ii) the html format (with a file name such as E8.html), which looks better on the World Wide Web; this is the format you will see when you download files from the data-base.
Several shell scripts are available for converting from the html format to other formats:
Each lattice is specified by some subset of the following properties
(further properties are being added):
For an explanation of (most of) these terms, see SPLAG:
"SPLAG" is an abbreviation for the book
Sphere Packings, Lattices and Groups,
by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988; 3rd edition in preparation, 1996;
Russian translation published by Mir, Moscow, 1990.
For a description of the Bacher polynomials, click here
Convention: 0 = FALSE, 1 = TRUE
(But click here to see the html version, which looks nicer!)
%NAME (obligatory) E8 %DIM (obligatory) 8 %BASIS (optional) 8 8 (no of rows, no of columns) then 64 numbers giving entries in generator matrix %TRIANGULAR_BASIS (optional) 8 8 (no of rows, no of columns) then 64 numbers giving entries in triangular and square generator matrix %GRAM (optional) 8 8 then 64 numbers giving entries of Gram matrix or %GRAM (optional) 8 0 then 8*9/2 = 36 numbers giving entries of Gram matrix in triangular format %DET (optional) 1 %GLUE_VECTORS (optional) nogl then nogl numbers for the glue vectors (including the 0 glue word) %PROPERTIES (optional) INTEGRAL=1 (means this is an integral lattice) %MINIMAL_NORM (optional) 2 (minimal norm) %KISSING_NUMBER (optional) 240 (kissing number, i.e. number of minimal vectors) %DENSITY (optional) .0625 (center density delta, Ref. SPLAG p. 13) %HERM (optional) 2 (Hermite number = 4 * delta^(2/dim) ) %NOTES (optional) comment lines go here %REFERENCES SPLAG p. 120. %LAST_LINE (obligatory last line)