Rush, J. A. A lower bound on packing density. (English) Zbl 0659.10033 Invent. Math. 98, No. 3, 499-509 (1989). If \(H\subseteq {\mathbb{R}}^ n\) is a bounded, convex body which is symmetric through each of the coordinate hyperplanes, then there exist codes which give rise, via Construction A of Leech and Sloane to lattice-packings of H whose density \(\Delta\) satisfies the logarithmic Minkowski-Hlawka bound, \(\liminf_{n\to \infty}\log_ 2^ n\sqrt{\Delta}\geq -1.\) This follows as a corollary of our main result, Theorem 9, a general way of obtaining lower bounds on the lattice-packing densities of various bodies. Unfortunately, when n is at all large, it is computationally prohibitive (although theoretically possible) to exhibit the arrangements explicitly. Reviewer: J.A.Rush Cited in 17 Documents MSC: 11H31 Lattice packing and covering (number-theoretic aspects) 52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry) 11H06 Lattices and convex bodies (number-theoretic aspects) Keywords:convex body; construction A; logarithmic Minkowski-Hlawka bound; lower bounds; lattice-packing densities PDF BibTeX XML Cite \textit{J. A. Rush}, Invent. Math. 98, No. 3, 499--509 (1989; Zbl 0659.10033) Full Text: DOI EuDML References: [1] Cassels, J.W.S.: An Introduction to the Geometry of Numbers. New York, Berlin, Heidelberg: Springer, second printing, 1971 · Zbl 0209.34401 [2] Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. New York, Berlin, Heidelberg: Springer, 1987 · Zbl 0634.52002 [3] Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. North-Holland Amsterdam: Elsevier, 1987 (This is an updated version of [7].) [4] Hilbert, D.: Mathematische Probleme. Arch. Math. Phys. 3rd ser.1, 44-63, 213-237 (1901) · JFM 32.0084.05 [5] Hlawka, E.: Zur Geometrie der Zahlen. Math. Z.49, 285-312 (1943) · Zbl 0028.20606 · doi:10.1007/BF01174201 [6] Leech, J., Sloane, N.J.A. Sphere packing and error-correcting codes. Can. J. Math.23, 718-745 (1971) · Zbl 0218.52008 · doi:10.4153/CJM-1971-081-3 [7] Lekkerkerker, C.G.: Geometry of Numbers. Groningen: Wolters-Noordhoff, 1969 [8] Litsin, S.N., Tsfasman, M.A.: Algebraic-geometric and number-theoretic packings of spheres (in Russian). Uspekhi Mat. Nauk40, 185-186 (1985) [9] Mac Williams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam: Elsevier, 2nd printing, 1978 [10] Minkowki, H.: Geometrie der Zahlen. I. Leipzig: B.G. Teubner, 1896 [11] Minkowski, H.: Gesammelte Abhandlungen, Chelsea, N.Y. (reprint), 1969 [12] Rogers, C.A.: Packing and Covering. University Press. Cambridge 1964 [13] Rogers, C.A.: Existence Theorems in the Geometry of Numbers. Ann. Math.48, 994-1002 (1947) · Zbl 0036.02701 · doi:10.2307/1969390 [14] Rush, J.A., Sloane, N.J.A.: An improvement to the Minkowski-Hlawka bound for packing superballs. Mathematika34, 8-18 (1987) · Zbl 0606.10028 · doi:10.1112/S0025579300013231 [15] Sloane, N.J.A.: Self-dual codes and lattices, in Relations Between Combinatorics and Other Parts of Mathematics. Proc. Symp. Pure Math.34, 273-308 (1979) · Zbl 0407.94009 [16] Sloane, N.J.A.: Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods.Comtemp. Math. 9, 153-185 (1982) · Zbl 0491.94024 [17] Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications. New York, Berlin, Heidelberg: Springer Vol I, 1985 and Vol II, 1988 · Zbl 0574.10029 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.