New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions

New upper bounds are given for the maximum number, 7m , of nonoverlapping unit spheres that can touch a unit sphere in n-dimensional Euclidean space, for n < 24. In particular it is shown that 78 = 240 and ~~~ = 196560. The problem of finding the maximum number, 7S , of billiard balls that can touch another billiard ball has a long and fascinating history (see [2]); the answer is known to be 12. But up to now no corresponding numbers TV have been determined for higher dimensions. We shah use the following theorem. THEOREM. Assume n > 3. Iff(t) is a real polynomial which satisfies (Cl) f(t) < Ofor-1 < t < 4, and (C2) the coeficients in the expansion off(t) in terms of Jacobipolynomials [I, chap. 22J This theorem may be found (implicitly or explicitly) in [3, 4, 61, but for completeness we sketch a simplified proof. A sphericaZ code C is any finite 210