Qn: Reduced Resolvable Design of Stage n and its Uniform Design

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

Reduced Resolvable Design of Stage n and its Uniform Design

Description

Builds the reduced resolvable design Q^*_n of Boudraa et al. (2013) directly, together with its associated uniform design U(p^m, (p^n)^{r^{**}_n}): the treatments are the p^m points of the affine geometry AG(m, p), each factor corresponds to one (m-n)-dimensional linear subspace W of GF(p)^m (one parallel class of affine flats), and the level of a treatment x on factor W is the coset x + W.

Usage

Qn(m, n, p = 2)

Arguments

m	Dimension of the projective geometry (an integer, m >= 2).
n	Stage of the recursion, an integer with 1 <= n <= m - 1.
p	Order of the Galois field GF(p); must be prime. Defaults to p = 2.

Details

The number of factors is the Gaussian binomial coefficient \binom{m}{m-n}_p. Any two distinct runs coincide in exactly \binom{m-1}{m-n-1}_p factors (the number of (m-n)-dimensional subspaces containing a fixed nonzero vector), so the design is equidistant; consequently it attains the discrete-discrepancy lower bound of Fang et al. (2004). Across stages the levels refine: if W' \subset W then the level partition induced by W is a coarsening of the one induced by W'.

For n = 1 the design is the saturated Rao-Hamming orthogonal array OA(p^m, (p^m-1)/(p-1), p, 2). For p = 2, n = 1, recoding its two levels to \pm 1 gives the Sylvester-Hadamard member of the Plackett-Burman class of order 2^m, up to row, column and level equivalence.

Subspaces are enumerated by canonical reduced row echelon form, which generates each subspace exactly once rather than scanning subsets of the p^m - 1 nonzero vectors. Labelling then visits, for every subspace and every run, the p^{m-n} elements of a coset, so the cost is approximately O\!\left(\binom{m}{m-n}_p \, p^m \, p^{m-n}\right) plus factor-construction overhead, with output size O\!\left(\binom{m}{m-n}_p \, p^m\right). This is why stages with similar factor counts can differ in runtime. On the machine used for the package benchmarks, Qn(5, 2) completed in well under a second, Qn(7, 1) in a few seconds and Qn(7, 3) (11811 factors) in a few minutes; timings are hardware dependent.

Value

A list with components:

V

Number of treatments (runs), p^m.

B

Number of blocks of Q^*_n.

R

Replication of each treatment (= number of factors).

K

Block size, p^{m-n}.

Lambda

Concurrence parameter (constant, see Details).

Levels

Number of levels of each factor, p^n.

UD

The uniform design: a p^m \times R matrix of levels 1..Levels.

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.

K.T. Fang, X. Lu, Y. Tang and J. Yin (2004). Constructions of uniform designs by using resolvable packings and coverings. Discrete Mathematics, 274, 25–40.

Examples

Q <- Qn(3, 1)            # stage 1 of PG(3,2): U(8, 2^7), Plackett-Burman
Q$UD

Q2 <- Qn(3, 2)           # stage 2: U(8, (2^2)^7), Example 3 of the paper
Q32 <- Qn(3, 1, p = 3)   # U(27, 3^13)
c(Q32$V, Q32$B, Q32$R, Q32$K, Q32$Lambda)

Qn(4, 2)$R               # 35 four-level factors on 16 runs

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