A general connection between a balanced fractional 2
m factorial (2
m-BFF) design of resolution 2
l+1 and a balanced array (B-array) of strength 2
l has been established by Yamamoto, Shirakura and Kuwada [11]. Recently, Kuwada [6] has obtained a connection between a 3
m-BFF design of resolution V and a B-array of strength 4.
As a generalization of a multidimensional partially balanced association scheme, a multidimensional relationship is considered here. By utilizing the property of a multidimensional relationship and its algebra, it can be proved that a necessary and sufficient condition for a fractional
sm factorial design
T of resolution 2
l+1 to be balanced is that
T is a B-array of strength 2
l with index set {λ
i0i1…
is-1}, provided the information matrix
MT is non-singular.
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