Constructive packings of cross polytopes

Rush, J. A.

doi:10.1112/s0025579300006719

Cited by 8 publications

(20 citation statements)

References 10 publications

(9 reference statements)

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“…A good lattice for the Rician channel is expected to be a dense packing lattice of crosspolytopes, i.e., a dense lattice with respect to the -norm. Construction of such lattices and their performance on the Rician channel are studied in [9], [2], [10].…”

Section: Observe That Wherementioning

confidence: 99%

Algebraic tools to build modulation schemes for fading channels

Giraud

Boutillon²,

Belfiore³

1997

IEEE Trans. Inform. Theory

274

243

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A unified framework is presented in order to build lattice constellations matched to both the Rayleigh fading channel and the Gaussian channel. The method encompasses the situations where the interleaving is done on the real components or on two-dimensional signals. In the latter case, a simple construction of lattices congruent to the densest binary lattices with respect to the Euclidean distance is proposed. It generalizes, in a sense to be clarified later, the structural construction proposed by Forney. These constellations are next combined with coset codes. The partitioning rules and the gain formula are similar to those used for the Gaussian channel.

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Section: Observe That Wherementioning

confidence: 99%

Algebraic tools to build modulation schemes for fading channels

Giraud

Boutillon²,

Belfiore³

1997

IEEE Trans. Inform. Theory

274

243

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“…Kabatjanskȋ and Levenšteȋn [11] derived from (14) (15) M (n, ϕ) ≤ (sin(ϕ/2)) −n 2 −(0.599+o(1))n , which holds for ϕ ≤ 63 0 and is the best asymptotic upper bound on M (n, ϕ). We note that if one uses the Blichfeldt gauge f n that comes from (15), then Blichfeldt's theorem yields the Kabatjanskȋ-Levenšteȋn upper bound (1) on δ(B n ). Together with (13), this indicates that the Blichfeldt method may provide a better asymptotic upper bound on δ(X n ) than the insphere volume ratio combined with (1).…”

Section: Discussionmentioning

confidence: 99%

“…In order to prove this we would need two things: an estimates on the o(n) term in (15) and asymptotic formulae for the intrinsic volumes V j (X n ) for all j = 0, . .…”

Section: Discussionmentioning

confidence: 99%

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The Packing Density of the $$n$$ n -Dimensional Cross-Polytope

Tóth

Fodor

Vígh

2015

Discrete Comput Geom

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Abstract. The packing density of the regular cross-polytope in Euclidean n-space is unknown except in dimensions 2 and 4 where it is 1. The only non-trivial upper bound is due to Gravel, Elser, and Kallus [9] who proved that for n = 3 the packing density of the regular octahedron is at most 1 − 1.4 . . . × 10 −12 . In this paper, we prove upper bounds for the packing density of the n-dimensional regular cross-polytope in the case that n ≥ 7. We use a modification of Blichfeldt's method [2] due to G. Fejes Tóth and W. Kuperberg [7].

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“…The Lee weight of (Xl,... ,Xn) in GF(p) n is understood to mean II II +'" "+ II II • We shall make use of Construction A, a method of using error-correcting codes to produce lattices, due to Leech and Sloane. See [9] or [3] about Construction A, and [13], [14], [15], [16], and [17] for some of its applications to packing and covering problems.…”

Section: Introduction Notation and Definitionsmentioning

confidence: 99%

An indexed set of density bounds on lattice packings

Rush

1994

Geom Dedicata

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Abstract. It has long been known that the admissibility of a lattice F with respect to a symmetric convex body B is equivalent to F being a packing lattice for ½ B. This fact is the basis of the interplay between the classical theory of the arithmetic minima of positive definite quadratic forms, on the one hand, and the dense lattice packing of spheres in R '~, on the other.We give an indexed set of bounds 6z(B) > aj, where 0 < j < n/2, on the lattice packing density of B. The case j = 0 reduces to the aforementioned long-known fact, and j = 1 was proved by Elkies, Odlyzko, and Rush, and was used to obtain record high packing densities for various superballs. The new cases make possible the use of smaller primes in the construction of these dense packings.Mathematics Subject Classification (1991): 11H31.

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Constructive packings of cross polytopes

Cited by 8 publications

References 10 publications

Algebraic tools to build modulation schemes for fading channels

Algebraic tools to build modulation schemes for fading channels

The Packing Density of the $$n$$ n -Dimensional Cross-Polytope

An indexed set of density bounds on lattice packings

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