  Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are
equivalent to weighted regular $t$-wise balanced designs, which generalize
combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as
well as weights. Partly motivated by the recent study on tight Euclidean
$t$-designs on two concentric spheres, in this paper we discuss tight relative
$t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild
condition that such a relative $t$-design induces the structure of a coherent
configuration with two fibers. Moreover, from this structure we deduce that a
polynomial from the family of the Hahn hypergeometric orthogonal polynomials
must have only integral simple zeros. The Terwilliger algebra is the main tool
to establish these results. By explicitly evaluating the behavior of the zeros
of the Hahn polynomials when they degenerate to the Hermite polynomials under
an appropriate limit process, we prove a theorem which gives a partial evidence
that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on
two shells are rare for large $t$.
