(I thank Avi Wigderson for telling me about it.) Serge Vlăduţ just arxived a paper with examples of lattices in such that the kissing number is exponential in . The existence of such a lattice is a very old open problem and the best record was an example where the kissing number is . (This is based on a 1959 construction of Barnes and Wall and can be found in the famous book by Conway and Sloane.) Noga Alon found in 1997 collections of spheres so that each one touches subexponentially many others. Vlăduţ’s result is very exciting!
A crucial step was a result by Ashikhmin, Barg, and Vlăduţ from 2001 showing that certain algebraic-geometry binary linear codes have exponentially many minimal weight vectors. (This result disproved a conjecture by Nati Linial and me.) This follows by applying in a very clever way early 80s constructions going back to Bos, Conway, Sloane and Barnes and Sloane, and this requires some subtle selection of the AG-codes that can be used. To quote the author: “In order to apply Constuctions D and E we need specific good curves (the curves in the Garcia Stichtenoth towers do not perfectly match our construction) and some Drinfeld modular curves perfectly suit our purposes.”
These lines of ideas and appropriate AG-codes look to me the most promising avenue towards exponential lower bound for the Borsuk problem. (See, e.g., this paper and this post) We need for that purpose AG codes X where the maximum weight occurs exponentially often (this need not be difficult) and that every subset of vectors that misses the maximum distance is exponentially smaller than |X|.