Resolvable: Resolvable Balanced Incomplete Block Design

In PGM2: Recursive Construction of Nested Resolvable Designs and Associated Uniform Designs over GF(p)

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)

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.

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.

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.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

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

PGM2 documentation built on Aug. 27, 2026, 9:07 a.m.