Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph

The Hamming graph H(n,q) is the whose vertices are words of length n over alphabet {0,1,…,q−1}, where two adjacent if they differ in exactly one coordinate. adjacency matrix has n+1 distinct eigenvalues n(q−1)−q⋅i with corresponding eigenspaces Ui(n,q) for 0≤i≤n. In this work we study functions belonging to a direct sum Ui(n,q)⊕Ui+1(n,q)⊕⋯⊕Uj(n,q) 0≤i≤j≤n. We find minimum cardinality support such q=2 and q=3, i+j>n. particular, eigenfunctions from eigenspace Ui(n,3) i>n2. Using correspondence between 1-perfect bitrades eigenvalue −1, size bitrade H(n,3).