Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: On Kissing Numbers in Dimensions 32 to 128. | On Kissing Numbers in Dimensions 32 to 128 Yves Edel Mathematisches Institut der Universitat Im Neuenheimer Feld 288 69120 Heidelberg Germany E. M. Rains and N. J. A. Sloane AT T Labs-Research 180 Park Avenue Florham Park NJ 07932-0971 USA Submitted April 9 1998 Accepted April 13 1998 ABSTRACT An elementary construction using binary codes gives new record kissing numbers in dimensions from 32 to 128. 1. Introduction Let Tn denote the maximal kissing number in dimension n that is the greatest number of n-dimensional spheres that can touch another sphere of the same size. Although asymptotic bounds on Tn are known 5 little is known about explicit constructions especially for n 32. Up to now the best explicit constructions have come from lattice packings. The kissing number T of the Barnes-Wall lattice1 BWn in dimension n 2m is 1 1 2 2 although for m 5 this is weak 146 880 9 694 080 and 1 260 230 400 in dimensions 32 64 and 128 for example . In contrast Quebbemann s lattice Q32 14 5 Chap. 8 has T 261 120. In recent years the kissing numbers of a few other lattices in dimensions 32 have been determined. Nebe 10 shows that the Mordell-Weil lattice MW44 has T 2 708 112. Nebe 11 shows that a 64-dimensional lattice constructed in 10 is extremal 3-modular and so by modular form theory has T 138 458 880. Bachoc and Nebe 1 give an 80-dimensional lattice with T 1 250 172 000. Elkies 6 calculated the kissing number of his lattice MW128 it is 218 044 170 240 over 170 times that of BW128. In the present note we show that an elementary construction using binary codes gives better values than all of these. However our packings are just local arrangements of spheres 1The subscript gives the dimension. THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R22 2 around the origin we do not know if they can be modified to produce dense infinite packings. 2. The construction Let C n d resp. C n d wỴ denote a set of binary vectors of length n and Hamming distance d apart resp. and with constant
