Cohn and Li (arXiv:2411.04916) improved the known lower bounds for the kissing number in dimensions 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. They prove a maximality statement only in dimension 17, and remark that it is unclear what the limits of such constructions might be. Ho subsequently improved dimension 19 by enlarging that code from 1024 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 the independent sets of an explicit Cayley graph on the 4096 words of the (24-n)-punctured extended Golay code, with connection set the nonzero words of weight less than ceil(n/4); hence the maximum size of the added-vector code is that graph's independence number. For n = 20 and n = 21 the graph is bipartite, so the independence number is exactly 2048, which is the number Cohn and Li already achieve. Their choice is therefore maximal: the bounds tau(20) >= 19448 and tau(21) >= 29768 cannot be improved by enlarging the added-vector code.
The bipartiteness has a short structural cause in the Steiner system S(5,8,24). Puncturing at a p-set P sends an octad to a word of weight 8 - |O intersect P|, and the block-intersection numbers lambda_5 = 1, lambda_4 = 5, lambda_3 = 21 determine the low-weight words exactly. For n = 21 there are 21 forbidden differences, all of odd weight 5, so total parity separates. For n = 20 there are 5 forbidden differences of weight 4; since two octads containing a common 4-set meet in precisely that set, their punctured images are pairwise disjoint and partition the 20 coordinates, so any transversal of the five blocks gives a separating functional. For n = 19 the forbidden differences comprise one word of weight 3 and twenty of weight 4, of mixed parity, and no separating functional exists: the graph is not bipartite. This explains why dimension 19 admitted Ho's improvement while dimensions 20 and 21 do not.
Scope. The result bounds the odd-sign construction, not the kissing number. It does not assert that tau(20) = 19448 or tau(21) = 29768; both remain open, and other constructions are not excluded. The ingredients are standard: the Steiner system S(5,8,24) and its intersection numbers, the duality of shortening and puncturing, and the existence of a perfect matching in a regular bipartite graph. The contribution is the reduction to an independence number, the observation that the resulting graph is bipartite precisely when n is not 19 among these three dimensions, and the resulting maximality statement, which answers for dimensions 20 and 21 a question the authors of the construction explicitly left open. Verification code accompanies the note; the certificates are small enough to check by hand.
Gonzalez, V. (2026). The added-vector code of the odd-sign construction is maximal in dimensions 20 and 21. Zenodo. https://doi.org/10.5281/zenodo.21702301