On the multidimensional permanent and $q$-ary designs

An $H(n,q,w,t)$ design is a collection of some $(n-w)$-faces of the hypercube $Q^n_q$ that perfectly pierce all $(n-t)$-faces $(n\geq w>t)$. An $A(n,q,w,t)$ design is a collection of some $(n-t)$-faces of $Q^n_q$ that perfectly cover all $(n-w)$-faces. The numbers of H-designs and A-designs are expressed in terms of the multidimensional permanent. Several constructions of H-designs and A-designs are given and the existence of $H(2^{t+1},s2^t,2^{t+1}-1,2^{t+1}-2)$ designs is proven for all $s,t\geq 1$.