Access vs. bandwidth in codes for storage

Maximum distance separable (MDS) codes are widely used in storage systems to protect against disks (nodes) failures. An (n, k, l) MDS code uses n nodes of capacity l to store k information nodes. The MDS property guarantees the resiliency to any n - k node failures. An optimal bandwidth (resp. optimal access) MDS code communicates (resp. accesses) the minimum amount of data during the recovery process of a single failed node. It was shown that this amount equals a fraction of 1/(n - k) of data stored in each node. In previous optimal bandwidth constructions, l scaled polynomially with k in codes with asymptotic rate <; 1. Moreover, in constructions with constant number of parities, i.e. rate approaches 1, l scaled exponentially w.r.t. k. In this paper we focus on the practical case of n - k = 2, and ask the following question: Given the capacity of a node l what is the largest (w.r.t. k) optimal bandwidth (resp. access) (k + 2, k, l) MDS code. We give an upper bound for the general case, and two tight bounds in the special cases of two important families of codes.