  In large scale distributed linear transform problems, coded computation plays
an important role to effectively deal with "stragglers" (distributed
computations that may get delayed due to few slow or faulty processors). We
propose a coded computation strategy, referred to as diagonal code, that
achieves the optimum recovery threshold and the optimum computation load. This
is the first code that simultaneously achieves two-fold optimality in coded
distributed linear transforms. Furthermore, by leveraging the idea from random
proposal graph theory, we design two random codes that can guarantee optimum
recovery threshold with high probability but with much less computation load.
These codes provide order-wise improvement over the state-of-the-art. Moreover,
the experimental results show significant improvement compared to both uncoded
and existing coding schemes.
