Outer bounds on the storage-repair bandwidth trade-off of exact-repair regenerating codes

In this paper three outer bounds on the storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set {(n, k, d), (α, β)} under the exact-repair (ER) setting are presented. The tradeoff under the functionalrepair (FR) setting was settled in the seminal work of Dimakis et al. that introduced the framework of regenerating codes as well as a subsequent paper by Wu. While it is known that the ER tradeoff coincides with the FR tradeoff at the extreme points of the tradeoff, known respectively as the minimum-storage-regenerating (MSR) and minimumbandwidth-regenerating (MBR) points, its characterization on the interior points remains open. The first outer bound presented here termed as the repair-matrix bound, in conjunction with a recent code construction known as improved layered codes characterizes the normalized ER tradeoff for the case of (n, k = 3, d = n−1). The repair-matrix bound is derived by building on top of the techniques introduced by Shah et al. and applies to every parameter set (n, k, d). It was earlier proved by Tian that the ER tradeoff lies strictly away from the FR tradeoff for the specific case (n = 4, k = 3, d = 3). The repair-matrix bound shows that a non-vanishing gap exists between the ER and FR tradeoffs for every parameter set (n, k, d). The second outer bound builds upon a bound due to Mohajer and Tandon and improves the bound using the very same techniques introduced in the Mohajer-Tandon paper and for this reason, is termed here as the improved Mohajer-Tandon bound. While for d = k the improved Mohajer-Tandon bound performs on par with the MohajerTandon bound, for d > k there is a significant improvement in the region of the tradeoff away from the MSR point. In the vicinity of the MSR point however, the repair-matrix bound outperforms the improved Mohajer-Tandon bound. In the third and final result, we restrict our focus to linear codes, and present an outer bound for the normalized ER tradeoff applicable to linear codes for the case k = d. In conjunction with the well-known class of layered codes, our third outer bound characterizes the normalized ER tradeoff in the case of linear codes for the case k = d = n−1. This bound is derived by analyzing the rank-structure of a parity-check matrix for a linear ER code. Index Terms Distributed storage; regenerating codes; exact-repair; storage-repair-bandwidth tradeoff; tradeoff characterization; outer bounds.