EN|RU Volume 21, No 6, 2014, P. 3-10 UDC 519.1 K. V. Vorob’ev and D. S. Krotov Bounds on the cardinality of a minimal 1-perfect bitrade in the hamming graph Abstract: 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. Keywords: Hamming graph, Krawtchouk polynomial, 1-perfect bitrade. Vorob’ev Konstantin Vasilievich 1,2 Krotov Denis Stanislavovich 1,2 1. S. L. Sobolev Institute of Mathematics, SB RAS, 4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia 2. Novosibirsk State University, 2 Pirogov St., 630090 Novosibirsk, Russia e-mail: vorobev@math.nsc.ru, krotov@math.nsc.ru © Sobolev Institute of Mathematics, 2015