New series of $2^{2m}$-dimensional universally strongly perfect lattices $Lambda_I $ and $Gamma_J $ are constructed with $$2BW_{2m} ^{#} subseteq Gamma _J subseteq BW_{2m} subseteq Lambda _I subseteq BW _{2m}^{#} .$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${mathcal U}_m:={mathcal C}_m (4^H_{bf 1}) $. The group ${mathcal U}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${bf F} _4$ containing ${bf 1}$, so the ring of polynomial invariants of ${mathcal U}_m$ is spanned by the genus-$m$ complete weight enumerators of such codes. This allows us to show that all the ${mathcal U}_m$ 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.
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