A new lower bound for the kissing number in 19 dimensions

We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. By the odd-sign construction of Cohn and Li, it is enough to find a binary code of length 19 and minimum distance 5 inside the ambient 5-punctured extended binary Golay code. We construct such a code explicitly, of size 1280. The construction is organized around a chain of linear codes $M\le K\le D$, $|M|=64$, $|K/M|=16$, and $|D/K|=4$. The 21 words of $D$ of weight 3 or 4 lie in exactly five nonzero $M$-cosets inside $K$. Those five cosets define a Cayley graph on $K/M\cong\mathbb F_2^4$ with connection set $\{e_1,e_2,e_3,e_4,e_1+e_2+e_3+e_4\}$, hence the Clebsch graph. A 5-coclique in that quotient lifts first to a 320-word code in $K$ and then, by taking all four cosets of $K$ in $D$, to the desired 1280-word code.