Minimum supports of functions on the Hamming graphs with spectral constraints

We study functions defined on the vertices of the Hamming graphs \(H(n,q)\). The adjacency matrix of \(H(n,q)\) has \(n+1\) distinct eigenvalues \(n(q-1)-q\cdot i\) with corresponding eigenspaces \(U_{i}(n,q)\) for \(0\leq i\leq n\). In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum \(U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)\) for \(0\leq i\leq j\leq n\). For the case \(n\geq i+j\) and \(q\geq 3\) we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case \(n