Kronecker product and reduced designs were defined by M. N. Vartak (1955). The main purpose of this paper is to give necessary and sufficient conditions for partially balanced incomplete block (PBIB) designs with $l$ associate classes constructed by the Kronecker product of some balanced incomplete block (BIB) designs to be reducible to PBIB designs with $l_1$ associate classes for a positive integer $l_1$ which is less than $l$.
Sanpei Kageyama. "On the Reduction of Associate Classes for Certain PBIB Designs." Ann. Math. Statist. 43 (5) 1528 - 1540, October, 1972. https://doi.org/10.1214/aoms/1177692385