In a factorial design several factors are controlled at two or more levels, and the effect on the response is investigated. There are two main types: Full Factorial Designs and Fractional Factorial Designs.
Used in experimental design. If the results of a Resolution III fractional factorial design leave questions unanswered it can be converted to a Resolution IV design by additional testing. The additional tests use the original design matrix but with all the signs reversed.
Fractional Factorial Designs 

Factorial designs involve testing several (n) factors at two levels, high and low. In a Fractional Factorial experiment only a half (2^{n1}), a quarter (2^{n2}), some other division by a power of two of the number of runs that would be required for a Full Factorial Experiment.
The example shows a 2^{31} design:
Run

A

B

C + AB

1

1

1

+1

2

+1

1

1

3

1

+1

1

4

+1

+1

+1

C is aliased with AB, thus it is a Resolution III design.
Factorial designs involve testing several factors at two levels, high and low. In a Full Factorial experiment every possible combination of factors and permutations is tested. If there are n factors this is 2^{n} levels.
The design matrix shows the combinations for a 2^{2} design
Run

A

B

AB

1

1

1

+1

2

+1

1

1

3

1

+1

1

4

+1

+1

+1

A type of design that allows blocking in two variables:

Operators (second blocking variable )

Material Batch (first blocking variable )

1

2

3

4

5

1

A

B

C

D

E

2

B

C

D

E

A

3

C

D

E

A

B

4

D

E

A

B

C

5

E

A

B

C

D

The treatments are represented by the Latin letters 'A', 'B' etc.
The design is restrictive because the number of classes of each blocking variable must equal the number of treatments. It is appropriate for eg. testing crops because the land can be organized into the appropriate number of 'blocks'.
These are Resolution III (screening designs). The number of runs in a PlackettBurman designs is a multiple of 4, thus avoiding the limitations of factorial and fractional factorial designs where the number of runs is 2^{k}.
The generating vectors for some PlackettBurman designs are below:
n=8 
(+ + +  +  ) 
n=12 
(+ +  + + +    + ) 
n=16 
(+ + + +  +  + +   +   ) 
n=20 
(+ +   + + + +  +  +     + + ) 
n=24 
(+ + + + +  +  + +   + +   +  +    ) 
n=36 
( +  + + +    + + + + +  + + +   +     +  +  + +  + ) 
The generating vectors have ‘n1’ rows. The last row of a PlackettBurman design contains only ‘‘ values.
The design will have 'n1' columns and a factor will be allocated to each column:
 put the generating vector into the first column (column ‘A’)
 copy the last value from column A into the first row of column B
 slide the rest of the values from A below that value
 copy the last value from column B into the first row of column C
 slide the rest of the values from column B below that value
 continue until all the columns have been populated
