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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2107.00958 |
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| _version_ | 1866909334762422272 |
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| author | Hollanti, Camilla Mantilla-Soler, Guillermo Miller, Niklas |
| author_facet | Hollanti, Camilla Mantilla-Soler, Guillermo Miller, Niklas |
| contents | It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice $\mathbb{Z}^n$ is the simplest example. The integer lattice is also an example of a generic well-rounded lattice, i.e., a well-rounded lattice with a minimal kissing number. However, the integer lattice has the worst density among well-rounded lattices. In this paper, the problem of constructing explicit generic well-rounded lattices with dense sphere packings is considered. To this end, so-called tame lattices recently introduced by Damir and Mantilla-Soler are utilized. Tame lattices came to be as a generalization of the ring of integers of certain abelian number fields. The sublattices of tame lattices constructed in this paper are shown to always result in either a generic well-rounded lattice or the lattice $A_n$, with density ranging between that of $\mathbb{Z}^n$ and $A_n$. In order to find generic well-rounded lattices with densities beyond that of $A_n$, explicit deformations of some known densest lattice packings are constructed, yielding a family of generic well-rounded lattices with densities arbitrarily close to the optimum. In addition to being an interesting mathematical problem on its own right, the constructions are also motivated from a more practical point of view. Namely, generic well-rounded lattices with high packing density make good candidates for lattice codes used in secure wireless communications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_00958 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Dense generic well-rounded lattices Hollanti, Camilla Mantilla-Soler, Guillermo Miller, Niklas Number Theory 11H31, 11F27, 11P21, 11H71 It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice $\mathbb{Z}^n$ is the simplest example. The integer lattice is also an example of a generic well-rounded lattice, i.e., a well-rounded lattice with a minimal kissing number. However, the integer lattice has the worst density among well-rounded lattices. In this paper, the problem of constructing explicit generic well-rounded lattices with dense sphere packings is considered. To this end, so-called tame lattices recently introduced by Damir and Mantilla-Soler are utilized. Tame lattices came to be as a generalization of the ring of integers of certain abelian number fields. The sublattices of tame lattices constructed in this paper are shown to always result in either a generic well-rounded lattice or the lattice $A_n$, with density ranging between that of $\mathbb{Z}^n$ and $A_n$. In order to find generic well-rounded lattices with densities beyond that of $A_n$, explicit deformations of some known densest lattice packings are constructed, yielding a family of generic well-rounded lattices with densities arbitrarily close to the optimum. In addition to being an interesting mathematical problem on its own right, the constructions are also motivated from a more practical point of view. Namely, generic well-rounded lattices with high packing density make good candidates for lattice codes used in secure wireless communications. |
| title | Dense generic well-rounded lattices |
| topic | Number Theory 11H31, 11F27, 11P21, 11H71 |
| url | https://arxiv.org/abs/2107.00958 |