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

Abstract

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.

On the Achievability Region of Regenerating Codes for Multiple Erasures

“…The convex hull, denoted by R, is the smallest convex set containing all known achievable points, obtained by all convex combinations (i.e., space-sharing) among the points achieved by Construction 1, described in (14), and also the MBCR point given by (4). The objective is therefore to determine which points are sufficient to describe R. We refer to these points as corner points of R. Figure 4 presents the achievable points for an (17,14,14,3) system. The achievable points of (13) are parameterized by r, such that e + 1 ≤ r ≤ e + k. For each r, we denote the corresponding point as (ᾱ r ,β r ).…”

“…We compute the normalized bandwidth, denoted byβ ′ r , achieved by the considered space-sharing, at the intermediate point α = α r , and then determine whetherβ ′ r >β r . Using (14) and (4), we obtain after simplification…”

“…Achievable points for an (n, k, d, e) =(17,14,14,3) system. The x-axis is the normalized storage per nodeᾱ and the y-axis is the normalized bandwidthβ.…”

Code constructions for multi-node exact repair in distributed storage

“…Substituting (20) in (2) and setting p = g − 2, we obtain (t + ge − 2e)ᾱ + (k − t − ge + 2e)β = (k − e)ᾱ + eβ = 1.…”

“…Later, code constructions for exact-repair interior points have been proposed in [12][13][14][15][16][17]. Moreover, there is a growing literature focused on understanding the fundamental limits of exact-repair regenerating codes [18][19][20][21], as opposed to the well-understood functional regenerating codes [1].…”