  Let the kissing number $K(d)$ be the maximum number of non-overlapping unit
balls in $\mathbb R^d$ that can touch a given unit ball. Determining or
estimating the number $K(d)$ has a long history, with the value of $K(3)$ being
the subject of a famous discussion between Gregory and Newton in 1694. We prove
that, as the dimension $d$ goes to infinity,
  $$
  K(d)\ge (1+o(1)){\frac{\sqrt{3\pi}}{4\sqrt2}}\,\log\frac{3}{2}\cdot
d^{3/2}\cdot \Big(\frac{2}{\sqrt{3}}\Big)^{d},
  $$ thus improving the previously best known bound of Jenssen, Joos and
Perkins by a factor of $\log(3/2)/\log(9/8)+o(1)=3.442...$. Our proof is based
on the novel approach from Jenssen, Joos and Perkins that uses the hard core
sphere model of an appropriate fugacity. Similar constant-factor improvements
in lower bounds are also obtained for general spherical codes, as well as for
the expected density of random sphere packings in the Euclidean space $\mathbb
R^d$.
