Berlekamp–van Lint–Seidel graph: existence at $(243,22,1,2)$

There exists a strongly regular graph with parameters $(243,22,1,2)$.

The Berlekamp–van Lint–Seidel graph is such a graph: its vertices are the $3^{11}/3^{6} = 243$ cosets of the perfect ternary Golay code $[11,6,5]_3$ in $\mathbb{F}_3^{11}$, with two cosets adjacent when their difference contains a vector of Hamming weight $1$. Equivalently it is the Cayley graph on $\mathbb{F}_3^{5}$ whose connection set consists of the $22$ syndromes $\pm h_i$ of the weight-one vectors, where $h_1, \dots, h_{11}$ are the columns of a parity check matrix of the code. It is the largest member of the family $\lambda = 1$, $\mu = 2$ known to exist, and the only known example that is not the trivial one on $9$ vertices.