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Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702-725] substitutedthe usual BLYM inequality for L-Sperner families with a set of M inequalitiesfor $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ type M-part Sperner families and showedthat if all inequalities hold with equality, then the family is homogeneous.Aydinian et al. [Australasian J. Comb. 48(2010), 133-141] observed that allinequalities hold with equality if and only if the transversal of the Spernerfamily 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 thispaper 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 thesemulti-families must be homogeneous with profile matrices that are strength M-kmixed orthogonal arrays. For k=M, homogeneity is not always true, but somenecessary conditions are given for certain simple families. Following themethods of Aydinian et al. [Australasian J. Comb. 48(2010), 133-141], we givenew constructions to simple mixed orthogonal arrays with constraint M, strengthM-k, using $m_i+1$ symbols in the ith column. We extend the convex hull methodto k-dimensional M-part Sperner multi-families, and allow additional conditionsproviding new results even for simple 1-part Sperner families.

Mixed orthogonal arrays, $k$-dimensional $M$-part Sperner multi-families, and full multi-transversals