Balanced nested designs and balanced n-ary designs

Balanced nested designs and balanced arrays

“…This method will be used here to provide several inÿnite series of symmetric balanced nested designs over ÿnite ÿelds. Similar results on pair-sum balanced nested designs have been reported in [13].…”

“…In [13], Fuji-Hara et al described a construction for pair-sum balanced nested designs from perpendicular arrays and nested designs. In this section we ÿrst propose a variation of this construction.…”

“…A generalization of Theorem 1.3 to balanced arrays of strength t is given by Fuji-Hara and Kuriki [12]. Recently, Fuji-Hara et al [13] deÿned another type of balanced nested designs, in which ij (x; y) + ji (x; y) is independent of x and y chosen for any distinct points belonging to V. Such a balanced nested design is called a pairsum balanced nested design in [13]. It is clear that if a balanced nested design is of symmetric type, then the design is also of pair-sum type.…”

“…It is clear that if a balanced nested design is of symmetric type, then the design is also of pair-sum type. It is shown [13] that the existence of a pair-sum balanced nested design satisfying some additional properties is equivalent to the existence of a balanced n-ary design. The notion of a balanced n-ary design was ÿrst introduced by Tocher [24].…”