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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 subgroupU  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.

Strongly perfect lattices sandwiched between Barnes–Wall lattices