Geometric constuction of balanced block designs with nested rows and columns.

*(English)*Zbl 0752.05006Consider 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 M. Singh and 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 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.

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 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.

Reviewer: G.L.Ebert (Newark / Delaware)

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\textit{R. Mukerjee} and \textit{S. Gupta}, Discrete Math. 91, No. 1, 105--108 (1991; Zbl 0752.05006)

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##### References:

[1] | Agrawal, H.L.; Prasad, J., Some methods of construction of balanced incomplete block designs with nested rows and columns, Biometrika, 69, 481-483, (1982) · Zbl 0494.62072 |

[2] | Cheng, C.S., A method for constructing balanced incomplete-block designs with nested rows and columns, Biometrika, 73, 695-700, (1986) · Zbl 0626.62075 |

[3] | Kurkjian, B.; Zelen, M., Applications of the calculus for factorial arrangements I. block and direct product designs, Biometrika, 50, 63-73, (1963) · Zbl 0114.35202 |

[4] | Preece, D.A., Nested balanced incomplete block designs, Biometrika, 54, 479-486, (1967) |

[5] | Raghavarao, D., Constructions and combinatorial problems in design of experiments, (1971), Wiley New York · Zbl 0222.62036 |

[6] | Singh, M.; Dey, A., Block designs with nested rows and columns, Biometrika, 66, 321-326, (1979) · Zbl 0407.62051 |

[7] | Street, D.J., Graeco-Latin and nested row and column designs, () · Zbl 0469.05016 |

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