Improved kissing numbers in seventeen through twenty-one dimensions
Henry Cohn, Anqi Li · arXiv (Cornell University) · 2024
Cohn and Li (arXiv:2411.04916) improved the known lower bounds for the kissing number indimensions 17 through 21 by an odd-sign construction whose final ingredient is a binary code,here called the added-vector code, chosen inside a punctured extended binary Golay code. Theyprove a maximality statement only in dimension 17, and remark that it is unclear what the limitsof such constructions might be. Ho subsequently improved dimension 19 by enlarging that code from1024 to 1280 words. Dimensions 20 and 21 had not been revisited. This note settles them. The admissible added-vector codes in dimension n are exactly theindependent sets of an explicit Cayley graph on the 4096 words of the (24-n)-punctured extendedGolay code, with connection set the nonzero words of weight less than ceil(n/4); hence the maximumsize of the added-vector code is that graph's independence number. For n = 20 and n = 21 the graphis bipartite, so the independence number is exactly 2048, which is the number Cohn and Li alreadyachieve. Their choice is therefore maximal: the bounds tau(20) >= 19448 and tau(21) >= 29768 cannotbe improved by enlarging the added-vector code. The bipartiteness has a short structural cause in the Steiner system S(5,8,24). Puncturing at ap-set P sends an octad to a word of weight 8 - |O intersect P|, and the block-intersection numberslambda_5 = 1, lambda_4 = 5, lambda_3 = 21 determine the low-weight words exactly. For n = 21 thereare 21 forbidden differences, all of odd weight 5, so total parity separates. For n = 20 there are5 forbidden differences of weight 4; since two octads containing a common 4-set meet in preciselythat set, their punctured images are pairwise disjoint and partition the 20 coordinates, so anytransversal of the five blocks gives a separating functional. For n = 19 the forbidden differencescomprise one word of weight 3 and twenty of weight 4, of mixed parity, and no separating functionalexists: the graph is not bipartite. This explains why dimension 19 admitted Ho's improvement whiledimensions 20 and 21 do not. Scope. The result bounds the odd-sign construction, not the kissing number. It does not assert thattau(20) = 19448 or tau(21) = 29768; both remain open, and other constructions are not excluded. Theingredients are standard: the Steiner system S(5,8,24) and its intersection numbers, the duality ofshortening and puncturing, and the existence of a perfect matching in a regular bipartite graph. Thecontribution is the reduction to an independence number, the observation that the resulting graph isbipartite precisely when n is not 19 among these three dimensions, and the resulting maximalitystatement, which answers for dimensions 20 and 21 a question the authors of the constructionexplicitly left open. Verification code accompanies the note; the certificates are small enough tocheck by hand.