Strongly perfect lattices sandwiched between Barnes–Wall lattices
Sihuang Hu, Gabriele Nebe · Journal of the London Mathematical Society · 2019
New series of 4 m -dimensional universally strongly perfect lattices Λ I and Γ J are constructed with 2 BW 2 m # ⊆ Γ J ⊆ BW 2 m ⊆ Λ I ⊆ BW 2 m # . The lattices are found by restricting the half-spin representations of the automorphism group of the Barnes–Wall lattice to its subgroup U m : = C m ( 4 1 H ) . The group U m is the Clifford–Weil group associated to the Hermitian self-dual codes over F 4 containing 1, so the ring of polynomial invariants of U m is spanned by the genus m complete weight enumerators of such codes. This allows us to show that all the U m -invariant lattices are universally strongly perfect. We introduce a new construction, D ( cyc ) , for chains of (extended) cyclic codes to obtain (bounds on) the minimum of the new lattices.