Mathematical Construction Methods - Automatic Generation of Combinatorial Test Data - page 21

2.2.1.2 Symbol-Collapsing

GivenaCA ( N , d 1 ··· d i ··· d k , t ) , if we replace a symbol v in the i th column with any

symbol in this set { 0 , 1 ,..., d i − 1 } other than v itself, we will get a CA ( N , d 1 ··· ( d i −

1 ) ··· d k , t ) .So

CAN ( d 1 ··· ( d i − 1 ) ··· d k , t ) ≤ CAN ( d 1 ··· d i ··· d k , t ).

2.2.1.3 Derivation

GivenaCA , if we select a symbol v from the i th column, then

extract the rows with a symbol v on the i th column, and finally delete the i th column,
we will get a CA

( N , d 1 ··· d i ··· d k , t )

( M , d 1 ··· d i − 1 · d i + 1 ··· d k , t − 1 ) , where M is no less than the

product of the t − 1 largest levels (excluding d i ). And we have

d i × CAN ( d 1 ··· d i − 1 · d i + 1 ··· d k , t − 1 ) ≤ CAN ( d 1 ··· d i ··· d k , t ).

2.2.1.4 Juxaposition

N , d 1 · N , d 1 · Given a CA ( d 2 ··· d k , t ) and a CA ( d 2 ··· d k , t ) , we can construct a

N + N ,( d 1 + d 1 ) ·

(

d 2 ··· d k ,

)

CA by relabeling the symbols in the first column

of one CA and putting it underneath the other.
In particular, we can construct a CA

t

( N , d 1 · d 2 ··· d k , t ) from a CA ( N , d 1 ·

d 2 ··· d k , t ) ,sowehave

CAN ( d 1 · d 2 ··· d k , t )) ≤ · CAN ( d 1 · d 2 ··· d k , t ).

5 2 4 1 3 2 2 7 For example, it is not difficult to find an instance of CA ( 25 , , 2 ) . Then

10 2 4 1 3 2 2 7

we can produce a CA ( 100 , , 2 ) by applying juxaposition twice. Obviously

this result is optimal.

2.2.2 Recursive Constructions

For many combinatorial objects, we can use mathematical results to produce large
objects from smaller ones. In this subsection, we briefly describe some of the results
for covering arrays.