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

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
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\textit{J. H. Conway} and \textit{N. J. A. Sloane}, J. Number Theory 48, No. 3, 373--382 (1994; Zbl 0810.11041)

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### Online Encyclopedia of Integer Sequences:

Norms of vectors in the a.c.c. lattice, divided by 2.Theta series of a.c.c. lattice.

Theta series for 10-dimensional 4-modular lattice Q10 with minimal norm 4.

Number of points in n-th shell of mcc lattice.

Value of s corresponding to norm of n-th shell of points in mcc lattice.

Value of t corresponding to norm of n-th shell of points in mcc lattice.