Strongly perfect lattices sandwiched between Barnes-Wall lattices

Abstract

New series of $2^{2m}$-dimensional universally strongly perfect lattices ${\Lambda }_I$ and ${\Gamma }_J$ are constructed with $$2B{W}_{2m}^{\#}\subseteq {\Gamma }_J\subseteq B{W}_{2m}\subseteq {\Lambda }_I\subseteq B{W}_{2m}^{\#}.$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${\mathcal{U}}_m:={\mathcal{C}}_m(4_1^H)$. The group ${\mathcal{U}}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${\mathbf{F}}_4$ containing $1$, 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.

Cited by in corpus (2)

• On Lattice Constructions D and D' from q-ary Linear Codes
• Extended Cyclic Codes Sandwiched Between Reed-Muller Codes