On lattices equivalent to their duals. (English) Zbl 0810.11041

Conway, J. H.; Sloane, N. J. A.

J. Number Theory 48, No. 3, 373-382 (1994).

The authors call a lattice in Euclidean \(n\)-space isodual if it is isometric to its dual. A lattice similar to its dual (e.g., any for \(n=2\)) can be rescaled to become isodual, and certain even lattices similar to their duals (interesting from the point of view of modular forms) hold the sphere packing records for \(n=4\), 8, 12, 16, 24, 32 and 48. The present paper, however, mainly deals with the case \(n=3\): using their version of the Selling parameters to describe cubic lattices, the authors find that a “mean” of the face-centered and body-centered cubic lattices (called the m.c.c. lattice) solves both the isodual packing and covering problems. The paper also lists the densest isodual lattices known for \(n= 5\), 6, 7, 9, 10, 11 and 14.

Reviewer: H.G.Quebbemann (Oldenburg)

MSC:

11H31	Lattice packing and covering (number-theoretic aspects)
11H06	Lattices and convex bodies (number-theoretic aspects)
52C07	Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

Keywords:

isodual covering; Selling parameters; cubic lattices; isodual packing; densest isodual lattices