3-Class Association Schemes and Hadamard Matrices of a Certain Block Form

We describe 3-class association schemes with adjacency matricesDisuch that for an appropriate numbering of the relations,D0 + D1 − D2 + D3is an Hadamard matrix of the following form:(JgH12···H1gH21Jg···H2g⋮⋮Hg1Hg2···Jg)whereg = 2sfor naturals, all blocks are of sizeg,Jgis the matrix andHijJg = JgHij = 0. e construct all Hadamard matrices of this form in which all blocksHijhave rank 1.