PGM2 (version 2.0.1)

Resolvable: Resolvable Balanced Incomplete Block Design

Description

Extracts the resolvable balanced incomplete block design (RBIBD) residual to block n of a BIBD constructed from a projective geometry: block n is deleted together with all its treatments, and the surviving parts of the remaining blocks form a resolvable design (for a BIBD from PG(m, p) this is the design of the affine geometry AG(m, p)).

Usage

Resolvable(n, mat)

Value

A list with components:

V

Number of treatments of the RBIBD, \(p^m\).

B

Number of blocks.

R

Replication of each treatment.

K

Size of each block, \(p^{m-1}\).

RBIB

The design: a matrix whose rows are the blocks.

Arguments

n

Index of the block (sub-variety) to be deleted; an integer between 1 and nrow(mat).

mat

The matrix of the BIBD (rows are blocks), e.g. the BIB component returned by BIB.

Author

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

Details

Works for any order p: residual block sizes are derived from the data instead of the binary-only formula used in PGM2 <= 1.2 (every remaining block loses exactly Lambda treatments to the deleted block, so all residual blocks have size K - Lambda). The blocks partition into R parallel classes of p blocks each.

References

A. Boudraa, Z. Gheribi-Aoulmi and M. Laib (2013). Recursive method for construction of resolvable nested designs and uniform designs associated. International Journal of Research and Reviews in Applied Sciences, 17(2), 167--176.

R.C. Bose (1942). A note on the resolvability of balanced incomplete block designs. Sankhya, 6, 105--110.

Examples

X <- BIB(4)                    # BIBD (31, 15, 7) from PG(4, 2)
Y <- Resolvable(1, X$BIB)      # RBIBD (16, 30, 15, 8, 7)
Y$V; Y$B; Y$K

Z <- BIB(2, p = 3)             # BIBD (13, 4, 1) from PG(2, 3)
W <- Resolvable(1, Z$BIB)      # RBIBD (9, 12, 4, 3, 1): AG(2, 3)
W$RBIB