A construction of integral lattices. (English) Zbl 0538.10028
Given an integral lattice L and a hyperbolic decomposition of some quotient L/pL, a technique for obtaining other lattices of the same dimension and discriminant as \(L\perp...\perp L\) is discussed. When applied to the \(D_ 4\) and \(E_ 8\) root lattices, this yields a new sphere-packing in \({\mathbb{R}}^{32}\), which is denser than those known up to now, and an extremal type II lattice in \({\mathbb{R}}^{64}\).
MSC:
| 11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |
| 11H06 | Lattices and convex bodies (number-theoretic aspects) |
| 11H31 | Lattice packing and covering (number-theoretic aspects) |
References:
| [1] | Sloane, Combinatorial Surveys (Proc. 6th British Combinatorial Conf pp 117– (1977) |
| [2] | Quebbemann, Mathematika 31 pp 12– (1984) |
| [3] | DOI: 10.2307/2007025 · Zbl 0502.52016 · doi:10.2307/2007025 |
| [4] | Leech, Canad. J. Math. 23 pp 718– (1971) · Zbl 0207.52205 · doi:10.4153/CJM-1971-081-3 |
| [5] | Eichler, Quadratische Formen und orthogonale Gruppen (1952) · doi:10.1007/978-3-662-01212-3 |
| [6] | Milnor, Symmetric bilinear forms (1973) · Zbl 0292.10016 · doi:10.1007/978-3-642-88330-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
