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Packing and Covering
- Publisher
- Wiley
- Copyright
- © 2022 London Mathematical Society.
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms.12699
- Publisher site
- See Article on Publisher Site
Abstract
Two constructions of lattice packings of n$ n$‐dimensional cross‐polytopes (ℓ1$ \ell _1$ balls) are described, the density of which exceeds that of any prior construction by a factor of at least 2nlnn(1+o(1))$ 2^{\frac{n}{\ln n}(1 + o(1))}$. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed–Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete cross‐polytopes of fixed radius r⩾3$ r \geqslant 3$ in Zn$ \mathbb {Z}^n$.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Dec 1, 2022