PGM2 (version 2.0.1)

Qn: 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)

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.

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.

Author

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

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.

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