History of Kissing numbers $\tau_n$

 - params:     n: '5'-  number:-    args:-    - '[40, 44]'-    state:-      lower:-      - '18'-      upper:-      - 1c+  number: '[40, 44]'   comment: $40$ is the kissing number of $D_5$ CITE{SPLAG}; the upper bound is the     semidefinite programming bound of Mittelmann and Vallentin CITE{MV}. - params:     n: '6'-  number:-    args:-    - '[72, 77]'-    state:-      lower:-      - '28'-      upper:-      - 2d+  number: '[72, 77]'   comment: $72$ is the kissing number of $E_6$ CITE{SPLAG}; the upper bound is by     de Laat, Leijenhorst and de Muinck Keizer CITE{dLLdMK}. - params:     n: '7'-  number:-    args:-    - '[126, 134]'-    state:-      lower:-      - 3u-      upper:-      - '46'+  number: '[126, 134]'   comment: $126$ is the kissing number of $E_7$ CITE{SPLAG}; the upper bound is by     Mittelmann and Vallentin CITE{MV}.

 - params:     n: '9'-  number:-    args:-    - '[306, 363]'-    state:-      lower:-      - 9i-      upper:-      - bb+  number: '[306, 363]'   comment: $306$ is attained by a nonlattice arrangement of Leech and Sloane CITE{LeechSloane};     the largest kissing number of a lattice in dimension $9$ is $272$, that of $\Lambda_9$

 - params:     n: '10'-  number:-    args:-    - '[510, 553]'-    state:-      lower:-      - fu-      upper:-      - h9+  number: '[510, 553]'   comment: $510$ is attained by an arrangement of Ganzhinov CITE{Ganzhinov}; the laminated     lattice $\Lambda_{10}$ has $336$ minimal vectors; the upper bound is by Machado

 - params:     n: '11'-  number:-    args:-    - '[604, 868]'-    state:-      lower:-      - is-      upper:-      - r4+  number: '[604, 868]'   comment: $604$ is attained by an arrangement found in 2026 by AI agents on the EinsteinArena     platform CITE{Bianchi}; $\Lambda_{11}$ has $438$ minimal vectors; the upper bound

 - params:     n: '12'-  number:-    args:-    - '[841, 1355]'-    state:-      lower:-      - q9-      upper:-      - 1ab+  number: '[841, 1355]'   comment: $841$ is attained by an arrangement of Takhanov, Assylbekov and Yun CITE{TAY},     one sphere more than the arrangements of size $840$ known before it; the Coxeter–Todd

 - params:     n: '13'-  number:-    args:-    - '[1154, 2064]'-    state:-      lower:-      - '142'-      upper:-      - 20g+  number: '[1154, 2064]'   comment: $1154$ is attained by an arrangement of Zinoviev and Ericson CITE{ZE};     $\Lambda_{13}$ has $906$ minimal vectors; the upper bound is by de Laat and Leijenhorst

 - params:     n: '14'-  number:-    args:-    - '[1932, 3174]'-    state:-      lower:-      - 1sc-      upper:-      - '336'+  number: '[1932, 3174]'   comment: $1932$ is attained by an arrangement of Ganzhinov CITE{Ganzhinov}; $\Lambda_{14}$     has $1422$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '15'-  number:-    args:-    - '[2564, 4853]'-    state:-      lower:-      - 2g4-      upper:-      - 4nl+  number: '[2564, 4853]'   comment: $2564$ is attained by a nonlattice arrangement of Leech and Sloane CITE{LeechSloane};     $\Lambda_{15}$ has $2340$ minimal vectors; the upper bound is by de Laat and Leijenhorst

 - params:     n: '16'-  number:-    args:-    - '[4320, 7320]'-    state:-      lower:-      - '470'-      upper:-      - 74o+  number: '[4320, 7320]'   comment: $4320$ is the kissing number of the Barnes–Wall lattice $\Lambda_{16}$     CITE{BarnesWall}; the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '17'-  number:-    args:-    - '[5730, 10978]'-    state:-      lower:-      - 5j2-      upper:-      - an2+  number: '[5730, 10978]'   comment: $5730$ is attained by an arrangement of Cohn and Li CITE{CohnLi}, obtained     by changing signs in the minimal vectors of a lattice; $\Lambda_{17}$ has $5346$

 - params:     n: '18'-  number:-    args:-    - '[7654, 16406]'-    state:-      lower:-      - 7f6-      upper:-      - g0m+  number: '[7654, 16406]'   comment: $7654$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{18}$     has $7398$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '19'-  number:-    args:-    - '[11948, 24417]'-    state:-      lower:-      - blc-      upper:-      - nr1+  number: '[11948, 24417]'   comment: $11948$ is attained by an arrangement of Ho CITE{Ho}, improving the $11692$     of Cohn and Li CITE{CohnLi}; $\Lambda_{19}$ has $10668$ minimal vectors; the upper

 - params:     n: '20'-  number:-    args:-    - '[19448, 36195]'-    state:-      lower:-      - ivo-      upper:-      - 13b3+  number: '[19448, 36195]'   comment: $19448$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{20}$     has $17400$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '21'-  number:-    args:-    - '[29768, 53524]'-    state:-      lower:-      - t28-      upper:-      - 1k8k+  number: '[29768, 53524]'   comment: $29768$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{21}$     has $27720$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '22'-  number:-    args:-    - '[49896, 80810]'-    state:-      lower:-      - 1gn8-      upper:-      - 2eta+  number: '[49896, 80810]'   comment: $49896$ is the kissing number of the laminated lattice $\Lambda_{22}$ CITE{Leech};     the upper bound is by de Laat and Leijenhorst CITE{dLL}. - params:     n: '23'-  number:-    args:-    - '[93150, 122351]'-    state:-      lower:-      - 2quu-      upper:-      - 3nff+  number: '[93150, 122351]'   comment: $93150$ is the kissing number of the laminated lattice $\Lambda_{23}$ CITE{Leech};     the upper bound is by de Laat and Leijenhorst CITE{dLL}.