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.