  The Hamming graph $H(n,q)$ is the graph whose vertices are the words of
length $n$ over the alphabet $\{0,1,\ldots,q-1\}$, where two vertices are
adjacent if they differ in exactly one coordinate. 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 study functions
belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus
U_j(n,q)$ for $0\leq i\leq j\leq n$. We find the minimum cardinality of the
support of such functions for $q=2$ and for $q=3$, $i+j>n$. In particular, we
find the minimum cardinality of the support of eigenfunctions from the
eigenspace $U_{i}(n,3)$ for $i>\frac{n}{2}$. Using the correspondence between
$1$-perfect bitrades and eigenfunctions with eigenvalue $-1$, we find the
minimum size of a $1$-perfect bitrade in the Hamming graph $H(n,3)$.
