  Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702-725] substituted
the usual BLYM inequality for L-Sperner families with a set of M inequalities
for $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ type M-part Sperner families and showed
that if all inequalities hold with equality, then the family is homogeneous.
Aydinian et al. [Australasian J. Comb. 48(2010), 133-141] observed that all
inequalities hold with equality if and only if the transversal of the Sperner
family corresponds to a simple mixed orthogonal array with constraint M,
strength M-1, using $m_i+1$ symbols in the $i^{\text{th}}$ column. In this
paper we define $k$-dimensional $M$-part Sperner multi-families with parameters
$L_P: P\in\binom{[M]}{k}$ and prove $\binom{M}{k}$ BLYM inequalities for them.
We show that if k<M and all inequalities hold with equality, then these
multi-families must be homogeneous with profile matrices that are strength M-k
mixed orthogonal arrays. For k=M, homogeneity is not always true, but some
necessary conditions are given for certain simple families. Following the
methods of Aydinian et al. [Australasian J. Comb. 48(2010), 133-141], we give
new constructions to simple mixed orthogonal arrays with constraint M, strength
M-k, using $m_i+1$ symbols in the ith column. We extend the convex hull method
to k-dimensional M-part Sperner multi-families, and allow additional conditions
providing new results even for simple 1-part Sperner families.
