New lower bounds on kissing numbers and spherical codes in high dimensions

Abstract: Let the kissing number K (d) be the maximum number of non-overlapping unit balls in Rᵈ 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) (1+o (1) ) {342}\, 32 d^3/2 (23) ^d, \ thus improving the previously best known bound of Jenssen, Joos and Perkins~On kissing numbers and spherical codes in high dimensions, Adv. Math. 335 (2018), 307--321 by a factor of (3/2) / (9/8) +o (1) =3. 442\,. Our proof is based on the novel approach from~that paper that uses the hard 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 Rᵈ.