## Showing papers on "Leech lattice published in 2017" ### Journal Article - **[The sphere packing problem in dimension 24](/content/papers/the-sphere-packing-problem-in-dimension-24-22fahscni8/index.html)** Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska Institutions: [Microsoft](/content/institutions/microsoft-2lvqci8u/index.html), [Stony Brook University](/content/institutions/stony-brook-university-2afaoni7/index.html), [Rutgers University](/content/institutions/rutgers-university-3kld8wdd/index.html), [Max Planck Society](/content/institutions/max-planck-society-3o0xx7lg/index.html), [Humboldt University of Berlin](/content/institutions/humboldt-university-of-berlin-1smin2jg/index.html) 01 May[2017](/content/journals/annals-of-mathematics-2xtq8btr/2017/index.html) - [Annals of Mathematics](/content/journals/annals-of-mathematics-2xtq8btr/index.html) TL;DR: In this article, it was shown that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. Abstract: Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case. ### Journal Article - **[Leech Constellations of Construction-A Lattices](/content/papers/leech-constellations-of-construction-a-lattices-1sy2kpfptz/index.html)** Nicola di Pietro, Joseph J. Boutros Institutions: [Texas A&M University at Qatar](/content/institutions/texas-a-m-university-at-qatar-fc7iephb/index.html) 01 Nov[2017](/content/journals/ieee-transactions-on-communications-r4vy07z3/2017/index.html) - [IEEE Transactions on Communications](/content/journals/ieee-transactions-on-communications-r4vy07z3/index.html) TL;DR: The problem of communicating over the additive white Gaussian noise (AWGN) channel with lattice codes is addressed and a new way to encode and demap Construction-A Voronoi lattices is presented. Abstract: The problem of communicating over the additive white Gaussian noise (AWGN) channel with lattice codes is addressed in this paper. Theoretically, Voronoi constellations have proved to yield very powerful lattice codes when the fine/coding lattice is AWGN-good and the coarse/shaping lattice has an optimal shaping gain. However, achieving Shannon capacity with these premises and practically implementable encoding algorithms is in general not an easy task. In this paper, a new way to encode and demap Construction-A Voronoi lattice codes is presented. These LDA lattice codes are based on dual-diagonal nonbinary low-density parity-check codes. With this choice, encoding, iterative decoding, and demapping have all linear complexity in the block length. ### Posted Content - **[On orbifold constructions associated with the Leech lattice vertex operator algebra](/content/papers/on-orbifold-constructions-associated-with-the-leech-lattice-3owwueawgo/index.html)** Ching Hung Lam, Hiroki Shimakura Institutions: [National Tsing Hua University](/content/institutions/national-tsing-hua-university-28827c61/index.html) 03 May[2017](/content/journals/arxiv-quantum-algebra-26ikgoz5/2017/index.html) - [arXiv: Quantum Algebra](/content/journals/arxiv-quantum-algebra-26ikgoz5/index.html) TL;DR: In this paper, it was shown that the structure of a strongly regular holomorphic vertex operator algebra of central charge $24$ is uniquely determined by its weight one Lie algebra if the Lie algebra has the type $A{3,4}^3A_{1,2}$, $A_{4,5}^2, $D{4,12}A_{2,6}. Abstract: In this article, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge $24$ is uniquely determined by its weight one Lie algebra if the Lie algebra has the type $A_{3,4}^3A_{1,2}$, $A_{4,5}^2$, $D_{4,12}A_{2,6}$, $A_{6,7}$, $A_{7,4}A_{1,1}^3$, $D_{5,8}A_{1,2}$ or $D_{6,5}A_{1,1}^2$ by using the reverse orbifold construction. ### Posted Content - **[On the Genus of the Moonshine Module](/content/papers/on-the-genus-of-the-moonshine-module-4kjfs0k3pm/index.html)** Gerald Höhn 20 Aug[2017](/content/journals/arxiv-quantum-algebra-26ikgoz5/2017/index.html) - [arXiv: Quantum Algebra](/content/journals/arxiv-quantum-algebra-26ikgoz5/index.html) TL;DR: In this paper, a simple description of Schellekens' seventy-one affine Kac-Moody structures of self-dual vertex operator algebras of central charge 24 by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with roots is provided. Abstract: We provide a novel and simple description of Schellekens' seventy-one affine Kac-Moody structures of self-dual vertex operator algebras of central charge 24 by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with roots. We also discuss a possible uniform construction procedure of the self-dual vertex operator algebras of central charge 24 starting from the Leech lattice. This also allows us to consider the uniqueness question for all non-trivial affine Kac-Moody structures. We finally discuss our description from a Lorentzian viewpoint. ### Journal Article - **[Holes of the Leech lattice and the projective models of K3 surfaces](/content/papers/holes-of-the-leech-lattice-and-the-projective-models-of-k3-5qob6fil8o/index.html)** Ichiro Shimada Institutions: [Hiroshima University](/content/institutions/hiroshima-university-27wvmnjq/index.html) 1 Feb 2017 TL;DR: Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, this article gave an effective bound for the set of isomorphism classes of projective models of fixed degree for certain K3 surfaces. Abstract: Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we provide an effective bound for the set of isomorphism classes of projective models of fixed degree for certain K3 surfaces. ### Journal Article - **[Construction of Holomorphic Vertex Operator Algebras of Central Charge 24 Using the Leech Lattice and Level p Lattices](/content/papers/construction-of-holomorphic-vertex-operator-algebras-of-2bnvzjtuku/index.html)** Ching Hung Lam, Hiroki Shimakura 1 Mar 2017 TL;DR: A more uniform construction of all 71 holomorphic vertex operator algebras in Schellekens' list using an idea proposed by G. Höhn is discussed in this paper. Abstract: In this article, we discuss a more uniform construction of all 71 holomorphic vertex operator algebras in Schellekens’ list using an idea proposed by G. Höhn. The main idea is to try to construct holomorphic vertex operator algebras of central charge 24 using some sublattices of the Leech lattice Λ and level p lattices. We study his approach and try to elucidate his ideas. As our main result, we prove that for an even unimodular lattice L and a prime order isometry g, the orbifold vertex operator algebra V ĝ Lg has group-like fusion. We also realize the construction proposed by Höhn for some special isometry of the Leech lattice of prime order. ### Posted Content - **[Half of an antipodal spherical design](/content/papers/half-of-an-antipodal-spherical-design-ypvv3excff/index.html)** Eiichi Bannai, Da Zhao, Lin Zhu, Yan Zhu, Yinfeng Zhu Institutions: [Shanghai Jiao Tong University](/content/institutions/shanghai-jiao-tong-university-1zokxin6/index.html) 29 Oct[2017](/content/journals/arxiv-combinatorics-11wv3snu/2017/index.html) - [arXiv: Combinatorics](/content/journals/arxiv-combinatorics-11wv3snu/index.html) TL;DR: In this paper, a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$ were studied, in particular root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$. Abstract: We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index T.