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Geometric constuction of balanced block designs with nested rows and columns. (English) Zbl 0752.05006
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 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.

##### MSC:
 05B05 Combinatorial aspects of block designs 62K10 Statistical block designs
##### Keywords:
balanced block designs; nested rows and columns
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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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