Bounds for the size of a minimal 1-perfect bitrade in a Hamming graph

We improve the available upper and lower bounds for the minimal size of the support of an eigenfunction of the Hamming graph H ( n, q ), where q > 2. In particular, the size of a minimal 1-perfect bitrade in H ( n, q ) is estimated. We show that the size of such a bitrade is at least 2 n −( n −1)/ q ( q − 2) ( n −1)/ q for q ≥ 4 and 3 n /2 (1 − O (1/ n )) for q = 3. Moreover, for n ≡ 1 mod q , where q is a prime power, we propose a construction of bitrades of size q ( q −2)( n −1)/ q 2 ( n −1)/ q +1 .