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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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