In this paper, we introduce a new subclass of $H$-matrices called $SD{D}_k$ matrices, which contains $SDD$ matrices and $SD{D}_1$ matrices as special cases, and present some properties of $SD{D}_k$ matrices. Based on these properties, we also provide new infinity norm bounds for the inverse of $SDD$ matrices and $SD{D}_k$ matrices. It is proved that these new bounds are better than some existing results in some cases. Numerical examples demonstrate the effectiveness of the obtained results.