The computations in the analysis of any equireplicate design can be carried out very easily if the number of treatments common to any two blocks is constant. A design with this property is called a Linked Block (LB) design and was introduced by Youden . It is well known that for a Balanced Incomplete Block (BIB) design to have a constant number of treatments in common between any two blocks, it is necessary and sufficient that it is symmetric, that is, the number of blocks is equal to the number of treatments. In this paper, necessary and sufficient conditions are derived for any design with a given treatment-structure matrix to be of the LB type and the results applied to Partially Balanced Incomplete Block (PBIB) designs. Finally a list is prepared of all LB designs in the class of two-associate PBIB designs enumerated by Bose, Shrikhande and Clatworthy .
"On Partially Balanced Linked Block Designs." Ann. Math. Statist. 28 (2) 488 - 493, June, 1957. https://doi.org/10.1214/aoms/1177706977