Collaborative Decoding of Polynomial Codes for Distributed Computation

We show that Polynomial codes (and some related codes) used for distributed matrix multiplication are interleaved Generalized Reed-Solomon codes and hence, can be collaboratively decoded. We consider a fault-tolerant setup where t out of N workers return erroneous values. For an additive random Gaussian error model, we show that for any $ t\leq N-K-1$, where K is the effective dimension of the code, all errors can be corrected with probability 1 while the decoding complexity is $O\left(\left(\frac{L}{L+1}\right)^{4}\left(N-K\right)^{4}+LN\right)$ for any $L\geq N-K-1$.