The Linear \(3\)-Cover of the Hermitean Forms Graph on \(81\) Vertices

The Hermitean forms graphs Her\((n,s)\) are a series of linear distance-regular graphs. The graph Her\((2,3)\) has the coset graph of the shortened ternary Golay code as an antipodal distance-regular cover. We give a new construction for this linear \(3\)-cover of \(Her\((2,3)\) and show that it is unique.