  We present a new $O(k^2 \binom{n}{k}^2)$ method for generating an orthogonal
basis of eigenvectors for the Johnson graph $J(n,k)$. Unlike standard methods
for computing a full eigenbasis of sparse symmetric matrices, the algorithm
presented here is non-iterative, and produces exact results under an
infinite-precision computation model. In addition, our method is highly
parallelizable; given access to unlimited parallel processors, the eigenbasis
can be constructed in only $O(n)$ time given n and k. We also present an
algorithm for computing projections onto the eigenspaces of $J(n,k)$ in
parallel time $O(n)$.
