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Lattices with exponentially large kissing numbers. (English) Zbl 1448.11124
The kissing number of a packing of equal spheres in Euclidean \(n\)-space is the maximal number of spheres touching another sphere in the packing. It is well known that the maximum kissing number \(\tau _n\) in \(n\) dimensions satisfies \[ 0.2 \leq \frac{\log_2(\tau _n) }{n} \leq 0.41 \] where the upper bound is by G. A. Kabatyanskiĭ and V. I. Levenshteĭn [Probl. Peredachi Inf. 14, No. 1, 3–25 (1978; Zbl 0407.52005)] and the lower bound comes from a random choice procedure due to various authors. For lattice packings there is no better upper bound known for \(\tau_n^{\ell }\) and the random argument giving the lower bound fails.
The present paper applies Construction D and Construction E to flags of algebraic geometric codes to prove the existence of lattices in certain dimensions \(n\) (e.g., \(n=5\cdot 2^{10a+2}, 3\cdot 2^{12a+3}, 7 \cdot 2^{14a+2}\), \(a\geq 2\)) with \(\frac{\log_2(\tau _n^{\ell }) }{n} \geq 0.03\) and therewith provides an exponential lower bound for the lattice kissing number.
This give an exponential lower bound in all dimensions and allows the author to show that that \(\lim \inf _{n\to \infty} (\frac{\log_2(\tau _n^{\ell }) }{n} ) \geq 0.02\).

MSC:
11H31 Lattice packing and covering (number-theoretic aspects)
11H71 Relations with coding theory
14G15 Finite ground fields in algebraic geometry
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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