Dense packings from quadratic fields and codes
The author starts with the integral ring \(\mathcal{O}_{K}\) of the field \(K= \mathbb{Q}(\sqrt{3})\). This means \(\mathcal{O}_{K}=\{u+\omega v:u,v\in \mathbb{Z\}}\) with \(\omega =(-1+\sqrt{3})/2\), and so \(\mathcal{O}_{K}\) corresponds to the densest packing of congruent circles in \(\mathbb{R}^{2}\). Then he considers certain prime ideals in \(\mathcal{O}_{K}\) and shows how they can be concatenated with codes to produce dense packings in \(\mathbb{R} ^{n}\). For several values of \(n\) he gets packings with the best known densities. Let \(\Delta _{n}\) denote the maximal density of packings of congruent balls in \(\mathbb{R}^{n}\). Then \(\lambda :=\lim \sup_{n\rightarrow \infty }\frac{1}{n}\log _{2}(\Delta _{n})\geq -1\). This is the well-known Minkowski bound. It is shown that this might be improved to \(\lambda \geq -0.8471\), if the best upper bound in coding theory developed by \textit{Y. Ben-Haim} and \textit{S. Litsyn} [Adv. Math. Commun. 1, No. 1, 83--92 (2007; Zbl 1198.94199)] could be achieved.
scientific article; zbMATH DE number 5321267