Serge Vlăduţ : Lattices with exponentially large kissing numbers

Conway and Sloane’s excellent tome, Sphere Packings, Lattices and Groups, is now due a fourth edition. Maryna Viakovska’s proofs of optimality of the E8 and Leech lattices should go in the fourth edition as well…

Combinatorics and more

   (I thank Avi Wigderson for telling me about it.) Serge Vlăduţ just arxived a paper with examples of lattices in $latex R^n$ such that the kissing number is exponential in $latex n$. The existence of such a lattice is a very old open problem and the best record was an example where the kissing number is $latex n^{log n}$. (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…

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