Steps: All Stages of the Recursive Construction

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

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)

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.

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).

Note

The return structure is tidier than in PGM2 <= 1.2, where the stages were spliced into one flat, partially unnamed list: each stage is now a proper sub-list.

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.

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)

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