The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum
Abstract
Let Λm be the Barnes-Wall lattice of dimension 2m, normalized to have minimum squared norm dm:=2m−1. We determine its complete norm spectrum strictly between dm and 2dm, proving that, for every integer m≥1, {|x|2:x∈Λm, dm<|x|2<2dm}={2dm−2j:j∈ℤ, ⌈ m/2⌉≤ j≤ m−2}. In particular, the norm spectrum in this interval is nonempty for m≥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.
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BibTeX
@misc{cao2026norm,
title = {{The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum}},
author = {Yuhang Cao and Sihuang Hu and Hexiang Huang and Mengrui Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15288},
eprint = {2610.15288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15288},
}
RIS
TY - UNPB
TI - The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum
AU - Yuhang Cao
AU - Sihuang Hu
AU - Hexiang Huang
AU - Mengrui Yang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15288
UR - https://arxiv.org/abs/2610.15288
AB - Let Λ_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≥1, {|x|^2:x∈Λ_m, d_m<|x|^2<2d_m}={2d_m−2^j:j∈ℤ, ⌈ m/2⌉≤ j≤ m−2}. In particular, the norm spectrum in this interval is nonempty for m≥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.
ER -