  In this paper, we investigate the randomized algorithms for block matrix
multiplication from random sampling perspective. Based on the A-optimal design
criterion, the optimal sampling probabilities and sampling block sizes are
obtained. To improve the practicability of the block sizes, two modified ones
with less computation cost are provided. With respect to the second one, a two
step algorithm is also devised. Moreover, the probability error bounds for the
proposed algorithms are given. Extensive numerical results show that our
methods outperform the existing one in the literature.
