Access versus bandwidth in codes for storage

Maximum distance separable (MDS) codes are widely used in storage systems to protect against disk (node) failures. A node is said to have capacity l over some field F{double-struck}, if it can store that amount of symbols of the field. 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 (respectively, optimal access) MDS code communicates (respectively, accesses) the minimum amount of data during the repair 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 when the asymptotic rate is less than 1. Moreover, in constructions with a constant number of parities, i.e., when the rate approaches 1, l is scaled exponentially with k. In this paper, we focus on the case of linear codes with linear repair operations and constant number of parities n-k=r, and ask the following question: given the capacity of a node l what is the largest number of information disks k in an optimal bandwidth (respectively, access) (k+r,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. The first is a family of codes with optimal update property, and the second is a family with optimal access property. Moreover, the bounds show that in some cases optimal-bandwidth codes have larger k than optimal-access codes, and therefore these two measures are not equivalent.