The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum

Let ${\mathrm{\Lambda }}_m$ be the Barnes-Wall lattice of dimension $2^m$, normalized to have minimum squared norm $d_m:=2^{m-1}$. We determine its complete norm spectrum strictly between $d_m$ and $2d_m$, proving that, for every integer $m\ge 1$, $\{\mathrm{\parallel }x{\mathrm{\parallel }}^2:x\in {\mathrm{\Lambda }}_m,d_m<\mathrm{\parallel }x{\mathrm{\parallel }}^2<2d_m\}=\{2d_m-2^j:j\in \mathbb{Z},\lceil m\mathrm{/}2\rceil \le j\le m-2\}$. In particular, the norm spectrum in this interval is nonempty for $m\ge 4$. The proof uses a Gaussian-integer model in which lattice automorphisms arising from Clifford unitaries transform a vector into a pair of coordinate subvectors in a lower-dimensional lattice. Divisibility of the differences of their squared norms and a matrix Parseval identity provide the arithmetic and averaging constraints needed for induction on $m$. Every listed norm is realized explicitly by a sum of two minimal vectors of the lattice.