Strongly perfect lattices sandwiched between Barnes-Wall lattices

TL;DR

This paper constructs new high-dimensional strongly perfect lattices using automorphism groups of Barnes-Wall lattices and Clifford-Weil groups, demonstrating their universal strong perfection and providing bounds on their minimum vectors.

Contribution

It introduces a novel construction of strongly perfect lattices via automorphism groups and invariant theory, expanding the class of known universally strongly perfect lattices.

Findings

Constructed new series of strongly perfect lattices in 2^{2m} dimensions.

Proved all these lattices are universally strongly perfect.

Developed a new cyclic code-based method to estimate lattice minima.

Abstract

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.