I am studying two lattices which seem to have kissing number 36 and 24, respectively, and I am curious if they are some known lattices in the literature.
I have looked up Conway and Sloane but didn't find any mentioning of such six-dimensional lattices. I notice that E6 lattice has kissing number 72 and its dual
$ E_6^* \equiv E_6 \cup ([1]+E_6) \cup ([2]+E_6), $
where $[1]$ and $[2]$ are two glue vectors, have kissing number 54. Is it possible to introduce more glue vectors and further reduce the kissing number to 36 or 24? Is there known relation between glue vectors and kissing numbers in six-dimensional or general lattices?
I have also looked up the catalog in Sloane's wegpage, but didn't find the lattices with these kissing numbers. If anyone has any idea, please let me know. Thanks!