Access Versus Bandwidth in Codes for Storage

Abstract

Maximum distance separable (MDS) codes are widely used in storage systems to protect against disk (node) failures. A node is said to have capacity <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> over some field F, if it can store that amount of symbols of the field. An ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k, l</i> ) MDS code uses n nodes of capacity l to store k information nodes. The MDS property guarantees the resiliency to any <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-k</i> 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/( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - k</i> ) of data stored in each node. In previous optimal bandwidth constructions, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> 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, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> is scaled exponentially with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> . In this paper, we focus on the case of linear codes with linear repair operations and constant number of parities <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> , 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) ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r, k, l</i> ) 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> than optimal-access codes, and therefore these two measures are not equivalent.