In this paper, an error bound for linear complementarity problems of strong $SDD$ ${}_1$ matrices is given. By properties of $SDD$ ${}_1$ matrices, a new subclass of $P$-matrices called $SD{D}_1$- $B$ is presented, which contains $B$-matrices. A new error bound of linear complementarity problems for $SD{D}_1$- $B$ is also provided, which improves the corresponding results. Numerical examples are given to illustrate the effectiveness of the obtained results.