On spherical codes generating the kissing number in dimensions 8 and 24
A spherical code is a subset of the unit sphere in \(\mathbb{R}^ n\). It is interesting to find a spherical code with many codewords given that two different words are at distance at least \(\rho\). The determination of the size of a maximum \(n\)-dimensional spherical code with \(\rho=1\) is equivalent to finding the kissing number \(\tau_ n\), that is, the largest possible number of equal size spheres that can touch a sphere in \(\mathbb{R}^ n\). The authors propose three methods (one of them well-known) for constructing a spherical code in \(\mathbb{R}^ n\) from binary error correcting codes. With a judicious choice of such binary codes, (and some additional work showing that the union of some constructed spherical codes still have large minimum distance), spherical codes with \(\tau_ 8\) and \(\tau_{24}\) words are constructed. As building blocks, the \([8,4,4]\) Hamming code and the \([24,12,8]\) Golay code are used.
scientific article; zbMATH DE number 94525