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

We describe two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) whose density exceeds that of the best prior construction by a factor of $ 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 $.