From: cet1@cus.cam.ac.uk (Chris Thompson)
Subject: Re: The Kissing numbers..
Date: 9 Oct 1999 18:55:52 GMT
Newsgroups: sci.math
Keywords: numbers of congruent spheres which touch each other in lattices

In article <37FE8353.B7CE4D4D@intel.com>,
Michael Jørgensen  <Michael.Joergensen@intel.com> wrote:
>
>Jorma Kypp| wrote:
>
>> I'm interest about the best sources to find out what has been
>> written about the kissing numbers?
>
>The first place to look is
>http://www.treasure-troves.com/math/KissingNumber.html

http://www.research.att.com/~njas/lattices/kiss.html contains some more
of the known bounds. [This is part of the Sloane-Nebe lattice database.]

Background reading should be Conway & Sloane "Sphere packings, lattices, 
and groups", 3rd edition (Springer) if you can afford it...

Chris Thompson
Email: cet1@cam.ac.uk
==============================================================================
   99c:94064 94B75
   Edel, Yves(D-HDBG); Rains, E. M.; Sloane, N. J. A.
   On kissing numbers in dimensions $32$ to $128$. (English. English
   summary)
   Electron. J. Combin. 5 (1998), no. 1, Research Paper 22, 5 pp.
   (electronic). [ORIGINAL ARTICLE] 
   
   Summary: "An elementary construction using binary codes gives new
   record kissing numbers in dimensions from 32 to 128."
   

   98m:52029 52C17 (05B40 11H31 94B75)  [same article as above]
   
   The authors give an easy construction that uses binary codes to define
   finite sets of vectors of equal length $l$ and minimal distance $\geq
   l$ in Euclidean $n$-space. They use lower bounds on the maximal number
   of codewords to show that their construction yields new records for
   the kissing number in dimension $n=32$, $36$, $40$, $44$, $64$, $80$
   and $128$.
   
                         Reviewed by Gabriele Nebe
     _________________________________________________________________
   98k:52048 52C17 (11H31 52-02)
   Zong, C.(4-LNDUC)
   The kissing numbers of convex bodies---a brief survey. 
   Bull. London Math. Soc. 30 (1998), no. 1, 1--10. [To journal home page]
   
   The translative kissing number $N(K)$ of a convex body $K$ in $\bold
   E\sp d$ is the largest number of non-overlapping translates of $K$
   which can be brought into contact with $K$ at its boundary. If the
   translates are taken from a lattice packing of $K$ we speak of the
   lattice kissing number $N\sp *(K)$ of $K$. The author surveys known
   results for $N(K)$, $N\sp *(K)$ and in part indicates the proofs:
   $N\sp *(K)\leq 3\sp d-1$ and $N\sp *(K)\leq 2\sp {d+1}-2$ for strictly
   convex $K$ (Minkowski); $N(K)\leq 3\sp d-1$ (Hadwiger, Groemer); $N\sp
   *(K)\geq d(d+1)$ (Swinnerton-Dyer); $18=N\sp *(T)\leq N(T)\leq 19$,
   $T$ a tetrahedron in $\bold E\sp 3$ (Zong); $N(B\sp d)\geq 2\sp
   {0.207d+o(d)}$, $B\sp d$ a ball in $\bold E\sp d$ (Wyner); $N(B\sp
   d)\leq\pi\sp {1/2}d\sp {3/2}2\sp {(d-1)/2}$, large $d$ (Rankin);
   $N(B\sp d)\leq 2\sp {0.401d+o(d)}$ (Kabatyanskii Levenshtein); $N\sp
   *(B\sp d)$, $d=4,\cdots,9$ (Watson); $N(B\sp d),\ d=3,8,24$ (Hoppe,
   Levenshtein, Odlyzko, Sloane); $N(K)=N\sp *(K)=6$, $K\subset \bold
   E\sp 2$ not a parallelogram (Grunbaum); $N(K)>N\sp *(K)$, $K$ suitable
   in $\bold E\sp d$, $d\geq 3$ (Zong).
   
                        Reviewed by Peter M. Gruber
     _________________________________________________________________
   95j:52035 52C17
   Boyvalenkov, Peter(BG-AOS)
   Small improvements of the upper bounds of the kissing numbers in
   dimensions $19$, $21$ and $23$. (English. English summary)
   Atti Sem. Mat. Fis. Univ. Modena 42 (1994), no. 1, 159--163.
   
   This is the third in a series of papers by the author devoted to
   developing a method of upper bounds for the kissing number $\tau\sb
   n$, which is the greatest number of unit spheres that can touch a
   given unit sphere in $n$-dimensional Euclidean space. The method is a
   refinement of the linear programming method introduced earlier [A. M.
   Odlyzko and N. J. A. Sloane, J. Combin. Theory Ser. A 26 (1979), no.
   2, 210--214; MR 81d:52010], and yields best possible bounds within a
   certain class of estimates described in the article. (The first two
   articles in the series are by the author [Serdica 18 (1992), no. 3-4,
   278--285; MR 94h:52039; J. Geom. 49 (1994), no. 1-2, 67--71; MR
   95e:52035].)
   
   The author reports the following improvements on the Odlyzko-Sloane
   bounds reported in the above reference: $\tau\sb {19}\leq 25\,901\to
   \leq 25\,900$, $\tau\sb {21}\leq 56\,852\to\leq 56\,851$, $\tau\sb
   {23}\leq128\,096\to\leq 128\,095$.
   
                        Reviewed by Robert M. Erdahl
   
   © Copyright American Mathematical Society 2000