Bose and Shimamoto [1] have shown that partially Balanced Incomplete Block (PBIB) designs are characterized by association schemes. But in the PBIB designs discussed so far by the various authors the relation of association is symmetrical or reflexive. The object of this paper is to study a class of useful designs in which the relation of association is not necessarily symmetrical. First the definition of this new design is given. The relation between the parameters and the number of independent secondary parameters are discussed next. Two types of association schemes are given defined as type I and type II, the latter being based on Galois Fields. For the general method of analysis, the model assumed is that of a design for two-way elimination of heterogeneity. The final equations arrived at are the same as those for PBIB designs. A numerical example illustrates the method still further. Two methods of construction, one each for type I and type II are given. The method for type II is based on the property of Galois Fields.