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

中文翻译：

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汉明图中的特征函数和最小 1-perfect bitrades

汉明图 $H(n,q)$ 是其顶点是字母表 $\{0,1,\ldots,q-1\}$ 上长度为 $n$ 的单词的图，其中两个顶点相邻，如果它们恰好在一个坐标上不同。$H(n,q)$ 的邻接矩阵有 $n+1$ 个不同的特征值 $n(q-1)-q\cdot i$ 和对应的特征空间 $U_{i}(n,q)$ for $0\ leq i\leq n$。在这项工作中，我们研究属于直接和 $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\ 的函数leq j\leq n$。我们找到对 $q=2$ 和 $q=3$，$i+j>n$ 支持此类函数的最小基数。特别地，我们从特征空间 $U_{i}(n,3)$ 中找到了对 $i>\frac{n}{2}$ 的特征函数支持的最小基数。使用 $1$-perfect bitrades 和具有特征值 $-1$ 的特征函数之间的对应关系，