Lattice Packings of Cross-polytopes from Reed-Solomon Codes and Sidon Sets

TL;DR

This paper introduces two new lattice packing constructions for n-dimensional cross-polytopes that significantly outperform previous methods in density, one using Reed-Solomon codes and the other based on Sidon sets, with implications for high-density packings.

Contribution

The paper presents novel lattice packing constructions for cross-polytopes with unprecedented density, utilizing Reed-Solomon codes and Sidon sets, advancing the state of the art in geometric packing.

Findings

Density exceeds previous constructions by a factor of at least 2^{n/ln n} as n→∞

Explicit lattice construction using Reed-Solomon codes

High asymptotic density for fixed radius r ≥ 3 in Z^n

Abstract

Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. 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 \geqslant 3 $ in $ \mathbb{Z}^n $.