PGM2: Recursive Construction of Nested Resolvable Designs and Associated Uniform Designs

Description

Implements recursive construction methods for balanced incomplete block designs (BIBDs), their second generation, resolvable BIBDs (RBIBDs), and uniform designs (UDs) derived from projective geometries over GF(2). It enables extraction of nested structures in multiple stages and supports recursive resolution processes, as introduced in Boudraa et al. (2013).

Note

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).

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

Maintainer: Mohamed Laib laib.med@gmail.com

References

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.

See Also

Useful links:

• https://mlaib.net

Examples

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)

Balanced Incomplete binary Blocks Designs

Description

Gives the configuration of a Balanced Incomplete binary Blocks Designs (BIBD) using a projective geometry on a Galois Field of order 2 GF(2).

Usage

BIB(m)

Arguments

Value

A LIST of:

1. [V] Number of treatments in the BIBD.

2. [B] Number of blocks of the BIBD.

3. [R] Repetition of each treatment.

4. [K] Size of each block.

5. [BIB] The configuration of the BIBD.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

References

D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.

Examples

m<-4
X<-BIB(m)   #BIB from PG(4,2)

Balanced Incomplete binary Blocks Designs of second Generation.

Description

Gives the configuration of a Balanced Incomplete binary Blocks Designs of seconde generation.

Usage

Gen(n, mat)

Arguments

Value

A LIST of:

1. [V] Number of treatments in the BIBD.

2. [B] Number of blocks of the BIBD.

3. [R] Repetition of each treatment.

4. [K] Size of each block.

5. [BIB] The configuration of the BIBD.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

References

D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.

Examples

## 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)

Resolvable Balanced Incomplete Block Design RBIBD

Description

Extracts the Resolvable Balanced Incomplete Block Design (RBIBD) of any BIBD constructed from a projective geometry.

Usage

Resolvable(n, mat)

Arguments

Value

A LIST of:

1. [V] Number of treatments in the RBIBD.

2. [B] Number of blocks of the RBIBD.

3. [R] Repetition of each treatment.

4. [K] Size of each block.

5. [BIB] The configuration of the RBIBD.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

References

D. Dugué Traité de statistique théorique et appliquée, Masson et Cie, 1958.

Examples

## 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)

Nested Resolvable Designs and associated Uniform Designs in different stages.

Description

Gives the different stages of nested design begening from a projective geometry, BIBD, BIBD seconde generation, RBIBD and associated uniform designs.

Usage

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

Arguments

Value

A LIST, with the parametrs and the configurations on each stages of recurrence.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

References

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

Examples

## 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)

Uniform designs

Description

Builds the uniform design (UD) using a Resolvable Balanced Incomplete Block Design (RBIBD)

Usage

Uniform(mat)

Arguments

Value

A LIST of:

1. [n] Number of experiments.

2. [F] The dimension of the design.

3. [UD] The configuration of the uniform design.

Author(s)

Mohamed Laib, Abla Boudraa and Zebida Gheribi-Aoulmi

References

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.

Examples

## 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)