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157
On Minkowski’s bound for latticepackings
, 2004
"... Abstract: We give a new proof of the MinkowskiHlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective. 1 1 ..."
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Abstract: We give a new proof of the MinkowskiHlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective. 1 1
Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Cited by 97 (1 self)
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Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1
and
"... The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪σ +... + ⎪ x n ⎪σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = pσ (p an odd prime) we exhibit a finite set of lattices, constructed fro ..."
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The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪σ +... + ⎪ x n ⎪σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = pσ (p an odd prime) we exhibit a finite set of lattices, constructed
and
"... The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪ σ +... + ⎪ x n ⎪ σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = p σ (p an odd prime) we exhibit a finite set of lattices, constructed ..."
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The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪ σ +... + ⎪ x n ⎪ σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = p σ (p an odd prime) we exhibit a finite set of lattices, constructed
and
"... The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪ σ +... + ⎪ x n ⎪ σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = p σ (p an odd prime) we exhibit a finite set of lattices, constructed ..."
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The MinkowskiHlawka bound implies that there exist lattice packings of ndimensional ‘‘superballs’ ’ ⎪ x 1 ⎪ σ +... + ⎪ x n ⎪ σ ≤ 1 (σ = 1, 2,...) having density Δ satisfying log 2 Δ ≥ − n ( 1 + o ( 1) ) as n → ∞. For each n = p σ (p an odd prime) we exhibit a finite set of lattices, constructed
A GILBERTVARSHAMOV TYPE BOUND FOR EUCLIDEAN PACKINGS.
"... Abstract. The present paper develops a method to obtain a GilbertVarshamov type bound for dense packings in the Euclidean spaces using suitable lattices. For the Leech lattice the obtained bounds are quite reasonable for large dimensions, better than the MinkowskiHlawka bound but not as good as th ..."
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Abstract. The present paper develops a method to obtain a GilbertVarshamov type bound for dense packings in the Euclidean spaces using suitable lattices. For the Leech lattice the obtained bounds are quite reasonable for large dimensions, better than the MinkowskiHlawka bound but not as good
Stability of Arakelov Bundles and Tensor Products without Global Sections
 DOCUMENTA MATH.
, 2003
"... This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that E ⊗F has at least a certain slope, but no global sections. It is motivated by an analogous theore ..."
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Cited by 2 (0 self)
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theorem of Faltings for vector bundles over algebraic curves and contains the MinkowskiHlawka theorem on sphere packings as a special case. The proof uses an adelic version of Siegel’s mean value formula.
Simultaneous packing and covering in the Euclidean plane
 Monatsh. Math
"... Dedicated to Professor Edmund Hlawka on the occasion of his 90th birthday In 1950, C.A. Rogers introduced and studied the simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, they have attracted the interests of many authors such as L. ..."
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Cited by 2 (0 self)
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. Fejes Tóth, S.S. Ry˘skov, G.L. Butler, K. Böröczky, H. Horváth, J. Linhart and M. Henk since, besides their own geometric significance, they are closely related to the packing densities and the covering densities of the convex body, especially to the MinkowskiHlawka theorem. However, so far our
On lower bounds of the density of packings of equal spheres in R^n
, 2003
"... We study lower bounds of the packing density of nonoverlapping equal spheres in R n, n � 2, as a function of the maximal circumradius of its Voronoi cells. Our viewpoint is that of Delone sets which allows to investigate the gap between the upper bounds of Rogers or KabatjanskiĭLeven˘stein and the ..."
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Cited by 5 (1 self)
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Leven˘stein and the MinkowskiHlawka type lower bounds for the density of latticepackings. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the BarnesWall, Craig and MordellWeil lattices, respectively BWn, A (r) n and MWn, and of the Delone constants of the BCH packings, when n
Radon Transforms and Packings
, 1999
"... We use some basic results and ideas from the integral geometry to study certain properties of group codes. The properties being studied are generalized weights and spectra of linear block codes over a finite field and their analogues for lattice sphere packings in Euclidean space. We do not obtain a ..."
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Cited by 3 (0 self)
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any new results about linear codes, although several short and simple proofs for known results are given. As to the lattices, we introduce a generalization of lattice Θfunctions, prove several identities on these functions, and prove generalizations of Siegel mean value and Minkowski–Hlawka theorems
Results 1  10
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157