It can be shown that the Hamming distance defines a positive semi-definite kernel (see e.g., here). However, is the kernel $$ K(\mathbf{x}, \mathbf{y}) = 1 - \frac{1}{n} \sum_{i=1}^n \mathbb{I}\{x_i = y_i\}$$ strictly positive-definite? That is, for any $n\in\mathbb{N}$, $c_1,\dots, c_n\in\mathbb{R}$ and $\mathbf{x}_1,\dots, \mathbf{x}_n\in\mathbb{R}^n$, $$ \sum_{i=1}^n \sum_{j=1}^n c_i c_j k(\mathbf{x}_i, \mathbf{x}_j) > 0. $$
Moreover, is the kernel defined by $\exp\{K(\mathbf{x}, \mathbf{y})\}$ strictly positive-definite?