PGM2 (version 2.0.1)

Steps: All Stages of the Recursive Construction

Description

Runs the whole recursion of Boudraa et al. (2013) starting from the projective geometry PG(m, p): at each stage the current BIBD yields a resolvable design (via Resolvable) and its associated uniform design (via Uniform), and the next-generation BIBD is extracted (via Gen) until the geometry is exhausted.

Usage

Steps(m, n, stage = "all", p = 2)

Value

A named list with (depending on stage) components

BIB1 (the first-generation BIBD as returned by

BIB), BIBg (a list of the next-generation BIBDs),

Resolvables (a list of the resolvable designs of every stage) and UDs (a list of the associated uniform designs).

Arguments

m

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

n

Index of the block (sub-variety) to be deleted at each stage. The same index is used at every stage, and the number of blocks shrinks along the recursion, so n must not exceed the block count of the last stage, \(p^2 + p + 1\) (the design of PG(2, p)).

stage

Stages wanted, a character vector (default "all"):

'S1'

The first-generation BIBD.

'S2'

The BIBDs of the following generations.

'S3'

The resolvable designs of every stage.

'S4'

The uniform designs associated with every stage.

p

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

Author

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.

Z. Gheribi-Aoulmi and M. Bousseboua (2005). Recursive methods for construction of balanced n-ary block designs. Serdica Mathematical Journal, 31, 189--200.

Examples

s <- Steps(3, 1)               # all stages of PG(3, 2)
names(s)
s$UDs[[1]]$UD                  # U(8, 2^7)

s3 <- Steps(3, 1, p = 3)       # all stages of PG(3, 3)
s3$UDs[[1]]$UD                 # U(27, 3^13)

Steps(4, 1, c('S1', 'S4'))     # first and last stage only, PG(4, 2)