Dualizing with respect to s-tuples
In a projective plane, if the roles of lines and points are interchanged, the dual geometry is obtained. Similar concept was introduced in a field of the design of experiments by Bose and Nair, who derived a new class of designs, by interchanging blocks and treatments in a given class of block designs. This concept of interchanging the roles of blocks and treatments is usually named as ``dualization''. We denote the dual of the design D by \(D^*_ 1\). This dualization, that is, writing the block numbers of blocks in which a treatment occurs in the original design, is extended to another concept as writing the block numbers of blocks in which a pair of treatments occurs in the originl design. This is named as ``dualization with respect to (w.r.t.) pairs'', which is denoted by \(D^*_ 2\) for a given block design D. In this note, the concept of dualization w.r.t. pairs is further generalized in the form as ``dualization w.r.t. s- tuples'' for \(s\geq 1\). This dual design is denoted by \(D^*_ s\). Applying this technique to certain designs yields new block designs \(D^*_ s\) for some values of s.
scientific article; zbMATH DE number 4019072