Bounds on the cardinality of a minimal $1$-perfect bitrade in the Hamming graph

We improve well-known upper and lower bounds on the minimal cardinality of the support of an eigenfunction of the Hamming graph $H(n,q)$ for $q>2$. In particular, the cardinality of a minimal $1$-perfect bitrade in $H(n,q)$ is estimated. We show that the cardinality of such bitrade is at least $2^{n-\frac{n-1}q}(q-2)^\frac{n-1}q$ in case $q\ge4$ and $3^\frac n2(1-O(1/n))$ in case $q=3$. Moreover, we propose a construction of bitrades of the cardinality $q^\frac{(q-2)(n-1)}q2^{\frac{n-1}q+1}$ for $n\equiv1\bmod q$ where $q$ is a prime power. Bibliogr. 10.