The model refers to a treatment design or to a block-treatment design in the presence of non-stochastic covariates, attached to each experimental unit. The problem is that of most efficient estimation of covariates parameters on one side and the treatment contrasts and/or block contrasts on the other side. If the components in the two sides are ''orthogonal'', then we can invoke optimality [in some sense] separately in each side and thereby characterize optimal designs in such a set-up.Following Lopes Troya (J. Statist. Plann. Inference 6 (1982a) 373, J. Statist. Plann. Inference 7 (1982b) 49), we investigate the underlying combinatorial problems in the context of CRD, RBD and BIBD in order to accommodate maximum number of covariates. Hadamard matrices and mutually orthogonal Latin squares play a central role in this study.