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