  Tensors are a fundamental operation in distributed and are commonly
distributed into multiple parallel tasks for large datasets. Stragglers and
other failures can severely impact the overall completion time. Recent works in
coded computing provide a novel strategy to mitigate stragglers with coded
tasks, with an objective of minimizing the number of tasks needed to recover
the overall result, known as the recovery threshold. However, we demonstrate
that this strict combinatorial definition does not directly optimize the
probability of failure.
  In this paper, we focus on the most likely event and measure the optimality
of a coding scheme more directly by its probability of decoding. Our
probabilistic approach leads us to a practical construction of random codes for
matrix multiplication, i.e., locally random alloy codes, which are optimal with
respect to the measures. Furthermore, the probabilistic approach allows us to
discover a surprising impossibility theorem about both random and deterministic
coded distributed tensors.
