Geometric constuction of balanced block designs with nested rows and columns (Q1179285)

Consider block designs with nested rows and columns of \(v\) varieties in \(b\) blocks of \(k=pq\) varieties each, arranged in \(p\) rows and \(q\) columns within each block with each variety replicated \(r\) times. Let \(N\), \(N_ 1\) and \(N_ 2\) be the variety-block, variety-row and variety-column incidence matrices, respectively. The design will be balanced if \(pN_ 1N^ t_ 1+qN^ t_ 2-NN^ t=gI+\lambda J\), for some integers \(g\) and \(\lambda\), where \(I\) is the identity matrix and \(J\) is the all ones matrix. Such a design is denoted by \(BIBRC\{v,b,r,p,q,\lambda\}\) if \(pq<v\) or by \(BCBRC\{v,b,r,p,q,\lambda\}\) if \(pq=v\). These designs were first introduced and discussed by \textit{M. Singh} and \textit{A. Dey} [Block designs with nested rows and columns, Biometrika 66, 321-326 (1979; Zbl 0407.62051)]. In the paper under review, the authors construct a \(BCBRC\) design with parameters \(v=s^ m\), \(b=r=\phi(m-1,t-1,s)\), \(p=s^ t\), \(q=s^{m-t}\) and \(\lambda=(s^{m-t}-1)\phi(m-2,t-2,s)\), where \(s\) is any prime power, \(m\geq 2\) is any integer, and \(t\) is any integer with \(1\leq t<m\). Here \(\phi(m,t,s)\) denotes the number of \(t\) flats in the projective space \(PG(m,s)\). The authors point out that if one also has a \(BIB\) design on \(s^ t\) varieties, then combining these two designs à la \textit{C.-S. Cheng} [A method for constructing balanced incomplete block designs with nested rows and columns, Biometrika 73, 695-700 (1986; Zbl 0626.62075)] will produce a \(BIBRC\) design.