The 3-class Association Schemes Connected with Checkered Hadamard Matrices of Order 16

In this paper we determine all symmetric and non-symmetric 3-class association schemes such that for their adjacency matrices Diwe have D0 + D3 = I4 ⊗ J4and D0 + D1 − D2 + D3is a checkered Hadamard matrix of order 16 (i.e. an Hadamard matrix consisting of 16 square blocks Hijof order 4 such that Hii = J4and HijJ4 = J4Hij = 0). It appears that there are, essentially, four symmetric and two non-symmetric 3-class association schemes of this kind.