Vartak  has considered the construction of experimental designs with the help of Kronecker products of matrices. The method is equivalent to the replacement of two elements, 0 and 1, by two matrices. A generalisation of the above idea is given by the author , using only the incidence matrices of balanced incomplete block (BIB) designs for substitution. In the present paper the same idea is extended to the case where substitution is by the incidence matrices of partially balanced incomplete block (PBIB) designs and factorial experiments. In Sections 2 and 3 some ideas regarding canonical vectors and PBIB designs are introduced. Section 4 deals with associable designs and their properties. In Section 5 balanced matrices are defined and in Section 6 a method is given for constructing designs by substituting for the elements of a balanced matrix, the incidence matrices of associable designs. The application of this method to the construction of factorial experiments is considered in Section 7.
"A Matrix Substitution Method of Constructing Partially Balanced Designs." Ann. Math. Statist. 31 (1) 34 - 42, March, 1960. https://doi.org/10.1214/aoms/1177705985