  Regenerating codes allow distributed storage systems to recover from the loss
of a storage node while transmitting the minimum possible amount of data across
the network. We present a systematic computer search for optimal systematic
regenerating codes. To search the space of potential codes, we reduce the
potential search space in several ways. We impose an additional symmetry
condition on codes that we consider. We specify codes in a simple alternative
way, using additional recovered coefficients rather than transmission
coefficients and place codes into equivalence classes to avoid redundant
checking. Our main finding is a few optimal systematic minimum storage
regenerating codes for $n=5$ and $k=3$, over several finite fields. No such
codes were previously known and the matching of the information theoretic
cut-set bound was an open problem.
