Correcting adversarial errors with generalized regenerating codes

Traditional regenerating codes are efficient tools to optimize both storage and repair bandwidth in storing data across a distributed storage system, particularly in comparison to erasure codes and data replication. In traditional regenerating codes, the collection of any k nodes can reconstruct all stored information and is called the reconstruction set, N-R. A failed node can be regenerated from any d surviving nodes. These collections of d nodes are called the regeneration sets, N-H. The number of reconstruction sets and the number of regeneration sets satisfy vertical bar N-R vertical bar = C-n(k) and vertical bar N-H vertical bar = C-n-1(d). In generalized regenerating codes, we will have, 1