On the density of lattice covering for $n=17$

M. M. Anzin

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 35-69.

Résumé

In present paper for $n=17$ improved estimate is obtained for the minimum density of lattice coverings of the Euclidean space with equal balls. This result is directed on a solution of a problem, known in the literature as “the problem of S. S. Ryshkov concerning lattice coverings” [1, 2]. This work is a continuation of a series of author's works. The work [3] is a basic work among them. Detailed definitions, the technique of the research and the proofs of the basic theorems are given there. We presume that the reader is acquainted with the results of the work [3]. The result based on a full description of the structure of the $L$-partition for the Coxeter lattice $A_{17}^6$ as well as the structure of the Voronoi–Dirichlet polyhedra as polyhedra defined by their vertices is given. On the basis of this description, exact value of the covering radius and the density function are evaluated for the lattice covering corresponding to this lattice. The values of the density function of the covering proved to be better (less) than the formerly known values. Thus, for $n=17$, improved estimate is obtained for the minimum density of lattice coverings of the Euclidean space with equal balls. Historically, the study of $L$-partitions of the Coxeter lattices $ A_n^r$ was initiated by S. S. Ryshkov in [4]. There are regular simplex $S$ relative volume 6 among $L$-body of the lattices $ A_n^r$ (named $F_1$ in table 1). It is well known from [4] $L$-body, which we use to start enumeration. Originally, we obtained $L$-bodies with a computer, using the well known «empty-ball method» of Delone (see [5]). As the first step of this method, we used the results of [4] for $S$. In the present paper, we complete the studies initiated in [4] for the form $A_{17}^6$. The similar results, earlier gotten by me for the dimentions $ n=11,\ldots,15$, were discussed in detail by me and S. S. Ryshkov at his lattice theory special seminars at the chair of discrete mathematics at MSU Faculty of Mechanics and Mathematics. Sergey Sergeyevich gave an appreciation for those results and named them «the results of physical and mathematical PhD's level», which was and continues to be a big stimulus for me to carry out new researches. The present result for $ n=17$ have surpassed all previous ones in a volume of calculations. I devote this result to the memory of my teacher — Sergey Sergeyevich Ryshkov. Bibliography: 19 titles.

Détail

Keywords: lattice, Coxeters lattices, lattice covering, covering density, $L$-body, $L$-partition.

Bibliographie

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