Let \(L_q(v)\) be the lattice of subsets of a given \(v\)-set or the lattice of the subspaces of a given \(v\)-dimensional vector space over \(\mathrm{GF}(q)\), in case \(q=1\) or \(q= \text{prime}\) power, respectively. Denote by \(X\) the vertex set of the lattice, by \(\le\) the ordering relation, by \(\wedge\) and \(\vee\) the meet and joint operators, respectively, and by \(\rho: X\to \{0,1,\ldots,v\}\) the rank function. The fibers \(X_0,X_1,\ldots,X_v\) of the lattice by definition are the spheres centered at the null of \(X\). The cardinality \(| X_k|\) is the \(q\)-binomial (or Gaussian) coefficient \({v\brack k}\). Let \(\Gamma \subset S^x\) denote the automorphism group of the lattice \(L_q(v)\). For any fixed integer \(n\), with \(1\le n\le v/2\), \(\Gamma\) acts on the fiber \(X_n\) as a transitive rank-\((n+1)\) permutation group and yields the partition \(\Theta = \{\theta_0,\theta_1,\ldots,\theta_n\}\) of \(X_n\times X_n\) into the \(n+1\) orbits \(\theta_i = \{(x,y)\in X_n\times X_n;\quad x\wedge y\in X_{n-i}\}\). Denote by \(p(i,j,s)\) the intersection numbers of the \(n\)-class association scheme \((X_n,\Theta)\): for given \((x,y)\in \theta_s\), the integer \(p(i,j,s)\) counts the points \(z\in X_ n\) such that \((x,z)\in \theta_ i\) and \((z,y)\in \theta_ i\) hold. In particular, we have \(p(i,j,0) = w(i)\delta_{i,j}\), where \(w(i)\) is the valency of the graph \((X_n,\theta_i)\) and is given by \(w(i)={n \brack i}[\frac{v-n}{i}]q^{i^ 2}\). Let \(m_k\) be defined by \(m_0=1\) and \(m_k = {v\brack k}-{v\brack k-1}\), \(k=1,2,\ldots,n\), and define a family \((Q_0(\omega),\ldots,Q_n(\omega))\) of orthogonal polynomials by requiring the following two properties: \(Q_k(\omega)\) is a polynomial of degree \(k\) with respect to the basic number \((1-q^{-\omega})/(1-q^{-1}),\) with \(Q_k(-\infty)>0\), and the \(Q_k(\omega)\) satisfy the orthogonality relations \[ \sum^{n}_{i=0}Q_ k(i)Q_{\ell}(i)w(i)={v\brack n} m_ k\delta_{k,\ell},\ 0\leq k, \ \ell \leq n, \] where the weight function \(w\) is given by the valencies. These conditions uniquely determine the \(Q_k(\omega)\). The \(Q_k(\omega)\) are given in classical \(q\)-hypergeometric series notation as \[ Q_k(\omega)=m_{k3}\phi_2(q^{\omega},q^k,q^{v+1-k}; q^n, q^{v-n}; q^{-1}; q^{-\omega -1}), \] and were originally introduced by \textit{W. Hahn} [Math. Nachr. 2, 4--34 (1949; Zbl 0031.39001)]. After studying the discrete harmonics related to the \(q\)-Hahn polynomials, the author applies his results to designs. Let \(Y\) be a nonempty subset of the fiber \(X_n\) and let \(t\) be an integer, \(0\le t\le n\). Then \(Y\) is called a \(t\)-design if it enjoys the following regularity property. For every given \(z\in X_t\), the number of points \(y\in Y\) with \(y\ge z\) is a constant (independent of \(z\)). The author proves (among other things) the following: Theorem. Let \(A(\omega)\) be a polynomial of degree \(\le n\) in the variable \((1-q^{-\omega})/(1-q^{-1})\) and define real numbers \(\alpha_k\) from the expansion \(A(\omega) = \alpha_0Q_0(\omega)+\alpha_1Q_1(\omega)+\cdots+\alpha_nQ_n(\omega)\) of \(A(\omega)\) in the basis of Hahn polynomials \(Q_k(\omega)\). Suppose \(A(i)\ge 0\) for \(i=0,1,\ldots,n\), with \(A(0)\ne 0\), and \(\alpha_ k\le 0\) for \(k>t\). Then the cardinality of any \(t\)-design \(Y\subset X_n\) is bounded by \(| Y| \ge A(0)/\alpha_0\).