| Title: | Nested Resolvable Designs and their Associated Uniform Designs |
|---|---|
| Description: | Construction method of nested resolvable designs from a projective geometry defined on Galois field of order 2. The obtained Resolvable designs are used to build uniform design. The presented results are based on <https://eudml.org/doc/219563> and A. Boudraa et al. (See references). |
| Authors: | Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi |
| Maintainer: | Mohamed Laib <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0-1 |
| Built: | 2025-01-26 04:25:02 UTC |
| Source: | https://github.com/mlaib/pgm2 |
Construction method of nested resolvable designs from a projective geometry defined on Galois field of order 2. The obtained Resolvable designs are used to build uniform design. The presented results are based on <https://eudml.org/doc/219563> and A. Boudraa et al. (See references).
This work established in same time with the article intitled: Abla Boudraa et al., Recursive method for construction of nested resolvable designs and uniform designs associated, International Journal of Research and Reviews in Applied Sciences, (17), Issue 2 (2013).
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
Maintainer: Mohamed Laib [email protected]
D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.
Gheribi-Aoulmi. Z and M. Bousseboua Recursive methods for construction of balanced n-ary block designs. Serdica Math.J (31), 2005,189-200
Fang.K.T et al., Constructions of uniform designs by using resolvable packings and coverings. Discrete Math. (19), 2003, 692-711.
Abla Boudraa, Zebida Gheribi-Aoulmi and Mohamed Laib. Recursive method for construction of nested resolvable designs and uniform designs associated. International Journal of Research and Reviews in Applied Sciences. Vol. 17, Issue 2 (2013).
Fang.K.T et al., Construction of uniform designs via super-simple resolvable t-designs. Util. Math. (66).2004, 15-32.
m<-4 X<-BIB(m) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) #Extract the RBIB n<-1 mat<-X$BIB X2<-Gen(n,mat) #Extract the BIBD of the second generation ## Not run: #Algorithm of the 3rd example in the paper : (Abla Boudraa & al) IJRRAS. #(17), Issue 2 (2013). bib<-BIB(3)$BIB mat<-NULL for(i in 1:15){mat[[i]]<-Gen(i,bib)$BIB2} x<-Reduce("rbind",mat) e<-dim(x)[1] b<-dim(x)[2] v<-bib[1,] for (i in 1:e) {for (j in 1:b) {if (any (x[i,j]==v)) {x[i,j]<-0}}} for (i in e:1) { if (all (x[i,]==0)) {x<-x[-i,]}} s<-x[1,] s<-s[s>0] h<-length(s) f<-dim(x)[1] x1<-matrix(nrow=f, ncol=h) for (i in 1:f) {x1[i,]<-x[i,][x[i,]>0]} A<-unique(x1) UD<-Uniform(A) ## End(Not run)m<-4 X<-BIB(m) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) #Extract the RBIB n<-1 mat<-X$BIB X2<-Gen(n,mat) #Extract the BIBD of the second generation ## Not run: #Algorithm of the 3rd example in the paper : (Abla Boudraa & al) IJRRAS. #(17), Issue 2 (2013). bib<-BIB(3)$BIB mat<-NULL for(i in 1:15){mat[[i]]<-Gen(i,bib)$BIB2} x<-Reduce("rbind",mat) e<-dim(x)[1] b<-dim(x)[2] v<-bib[1,] for (i in 1:e) {for (j in 1:b) {if (any (x[i,j]==v)) {x[i,j]<-0}}} for (i in e:1) { if (all (x[i,]==0)) {x<-x[-i,]}} s<-x[1,] s<-s[s>0] h<-length(s) f<-dim(x)[1] x1<-matrix(nrow=f, ncol=h) for (i in 1:f) {x1[i,]<-x[i,][x[i,]>0]} A<-unique(x1) UD<-Uniform(A) ## End(Not run)
Gives the configuration of a Balanced Incomplete binary Blocks Designs (BIBD) using a projective geometry on a Galois Field of order 2 GF(2).
BIB(m)BIB(m)
m |
Dimension of the projective geometry defined on GF(2) |
A LIST of:
V Number of treatments in the BIBD.
B Number of blocks of the BIBD.
R Repetition of each treatment.
K Size of each block.
BIB The configuration of the BIBD.
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.
m<-4 X<-BIB(m) #BIB from PG(4,2)m<-4 X<-BIB(m) #BIB from PG(4,2)
Gives the configuration of a Balanced Incomplete binary Blocks Designs of seconde generation.
Gen(n, mat)Gen(n, mat)
n |
The sub-variety of the block to be deleted. |
mat |
The matrix of the BIB. |
A LIST of:
V Number of treatments in the BIBD.
B Number of blocks of the BIBD.
R Repetition of each treatment.
K Size of each block.
BIB The configuration of the BIBD.
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.
## Not run: m<-4 X<-BIB(m) #BIB from PG(4,2) n<-1 mat<-X$BIB X2<-Gen(n,mat) #Extracts the BIB second generation ## End(Not run)## Not run: m<-4 X<-BIB(m) #BIB from PG(4,2) n<-1 mat<-X$BIB X2<-Gen(n,mat) #Extracts the BIB second generation ## End(Not run)
Extracts the Resolvable Balanced Incomplete Block Design (RBIBD) of any BIBD constructed from a projective geometry.
Resolvable(n, mat)Resolvable(n, mat)
n |
The sub-variety of the block to be deleted. |
mat |
The matrix of the BIB. |
A LIST of:
V Number of treatments in the RBIBD.
B Number of blocks of the RBIBD.
R Repetition of each treatment.
K Size of each block.
BIB The configuration of the RBIBD.
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.
## Not run: m<-4 X<-BIB(m) #BIBD from PG(4,2) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) #Extracts the RBIBD ## End(Not run)## Not run: m<-4 X<-BIB(m) #BIBD from PG(4,2) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) #Extracts the RBIBD ## End(Not run)
Gives the different stages of nested design begening from a projective geometry, BIBD, BIBD seconde generation, RBIBD and associated uniform designs.
Steps(m, n, stage = "all")Steps(m, n, stage = "all")
m |
Dimension of the projective geometry defined on GF(2). |
n |
The sub-variety of the block to be deleted. |
stage |
Stage of recurrence wanted (single value or vector) : 'all' Print all designs. 'S1' Print the configurations of BIB of first generation. 'S2' Print the configurations of BIB of seconde generation. 'S3' Print the configurations of RBIB. 'S4' Print the configurations of the uniform designs associate to each nested resolvable. |
A LIST, with the parametrs and the configurations on each stages of recurrence.
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
Gheribi-Aoulmi. Z and M. Bousseboua Recursive methods for construction of balanced n-ary block designs. Serdica Math.J (31), 2005,189-200
## Not run: list1<-Steps(4,1) #Get all stages : of the PG(4,2) list2<-Steps(4,1,c('S1','S2')) #Get the 2 first stages : of the PG(4,2) list3<-Steps(4,1,c('S1','S4')) #Get the first & the last stage : of the PG(4,2) list4<-Steps(4,1,'S4') #Get the last stage : of the PG(4,2) ## End(Not run)## Not run: list1<-Steps(4,1) #Get all stages : of the PG(4,2) list2<-Steps(4,1,c('S1','S2')) #Get the 2 first stages : of the PG(4,2) list3<-Steps(4,1,c('S1','S4')) #Get the first & the last stage : of the PG(4,2) list4<-Steps(4,1,'S4') #Get the last stage : of the PG(4,2) ## End(Not run)
Builds the uniform design (UD) using a Resolvable Balanced Incomplete Block Design (RBIBD)
Uniform(mat)Uniform(mat)
mat |
The matrix of the RBIBD. |
A LIST of:
n Number of experiments.
F The dimension of the design.
UD The configuration of the uniform design.
Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi
Fang.K.T et al., Constructions of uniform designs by using resolvable packings and coverings. Discrete Math. (19), 2003, 692-711.
Fang.K.T et al., Construction of uniform designs via super-simple resolvable t-designs. Util. Math. (66).2004, 15-32.
## Not run: m<-4 X<-BIB(m) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) ud<-Uniform(Y$RBIB) #Uniform design ## End(Not run)## Not run: m<-4 X<-BIB(m) n<-1 mat<-X$BIB Y<-Resolvable(n,mat) ud<-Uniform(Y$RBIB) #Uniform design ## End(Not run)