PGM2 (version 2.0.1)

BIB: Balanced Incomplete Block Design from PG(m, p)

Description

Builds the symmetric balanced incomplete block design (BIBD) whose treatments are the points of the projective geometry PG(m, p) over the Galois field GF(p) and whose blocks are its hyperplanes.

Usage

BIB(m, p = 2)

Value

A list with components:

V

Number of treatments, \((p^{m+1}-1)/(p-1)\).

B

Number of blocks (equal to V: the design is symmetric).

R

Replication of each treatment, \((p^m-1)/(p-1)\).

K

Block size (equal to R).

Lambda

Concurrence parameter, \((p^{m-1}-1)/(p-1)\).

BIB

The design: a B x K matrix whose rows are the blocks, containing treatment labels.

Arguments

m

Dimension of the projective geometry (an integer, m >= 2).

p

Order of the Galois field GF(p); must be prime. Defaults to p = 2, which reproduces the designs of PGM2 <= 1.2 exactly.

Author

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

Details

Treatments are numbered by enumerating the canonical representatives of the projective points in base-p counting order. Block i consists of the points x lying on the hyperplane a_i . x = 0 (mod p), where a_i is the i-th point (point-hyperplane duality).

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.

D. Dugue (1958). Traite de statistique theorique et appliquee. Masson et Cie, Paris.

Examples

X <- BIB(4)        # BIBD (31, 15, 7) from PG(4, 2)
X$V; X$K; X$Lambda

Y <- BIB(2, p = 3) # BIBD (13, 4, 1): the projective plane of order 3
Y$BIB