# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  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  Cheng, C.S., A method for constructing balanced incomplete-block designs with nested rows and columns, Biometrika, 73, 695-700, (1986) · Zbl 0626.62075  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  Preece, D.A., Nested balanced incomplete block designs, Biometrika, 54, 479-486, (1967)  Raghavarao, D., Constructions and combinatorial problems in design of experiments, (1971), Wiley New York · Zbl 0222.62036  Singh, M.; Dey, A., Block designs with nested rows and columns, Biometrika, 66, 321-326, (1979) · Zbl 0407.62051  Street, D.J., Graeco-Latin and nested row and column designs, () · Zbl 0469.05016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.