Reed-Solomon码与Sidon集交叉多面体的格填充
Lattice Packings of Cross-polytopes from Reed-Solomon Codes and Sidon Sets
摘要：
描述了$ n $维交叉多面体（$ \ell_1 $球）的晶格填充的两种构造，当$ n \to \infty $时，其密度超过任何现有构造的密度至少$ 2^{\frac{n}{\ln n}(1 + o(1))} $倍。第一类格是显式的，并且通过将构造A应用于一类Reed-Solomon码而获得。第二个家族具有次指数构造复杂性，并且基于有限阿贝尔群中的Sidon集的概念。基于Sidon集的构造也给出了在$ \mathbb{Z}^n $中填充固定半径$ r \geqslant 3 $的离散交叉多面体的已知最高渐近密度.
英文摘要：
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 $.
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